The Burau representation of the braid group is faithful for n = 4

# The Burau representation is faithful for 𝑛=4
Source: [https://arxiv.org/html/2607.05283](https://arxiv.org/html/2607.05283)
###### Abstract

In this paper we use ideas introduced earlier by Moody\[[MOO91](https://arxiv.org/html/2607.05283#bib.bib8)\], Long\[[LON86](https://arxiv.org/html/2607.05283#bib.bib3)\], Long–Paton\[[LP93](https://arxiv.org/html/2607.05283#bib.bib4)\]and Bigelow\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10)\]to prove the theorem of the title, that the Burau representation of the classical braid groupB4\\operatorname\{B\}\_\{4\}is faithful\. An immediate corollary is that the Jones representation ofB4\\operatorname\{B\}\_\{4\}is also faithful\.

## 1Introduction

LetDnD\_\{n\}be a disk withnnmarked pointsp1,p2,…,pn\{p\_\{1\},p\_\{2\},\\dots,p\_\{n\}\}in its interior\. Thebraid groupBn\\operatorname\{B\}\_\{n\}will be interpreted in this paper as the mapping class groupMod⁡\(Dn\)\\operatorname\{Mod\}\(D\_\{n\}\)\. In 1935 Werner Burau introduced a representation ofBn\\operatorname\{B\}\_\{n\}intoGLn−1⁡\(ℤ​\[t,t−1\]\)\\operatorname\{GL\}\_\{n\-1\}\(\\mathbb\{Z\}\[t,t^\{\-1\}\]\), now known as the reduced Burau representation\[[BUR35](https://arxiv.org/html/2607.05283#bib.bib5)\]\. We will work mainly with the unreduced Burau representationρn:Bn→GLn⁡\(ℤ​\[t,t−1\]\)\\rho\_\{n\}:\\operatorname\{B\}\_\{n\}\\to\\operatorname\{GL\}\_\{n\}\(\\mathbb\{Z\}\[t,t^\{\-1\}\]\), and henceforth we refer toρn\\rho\_\{n\}simply as the Burau representation\.

Magnus–Peluso first established the faithfulness of the Burau representation forn=3n=3by a direct algebraic calculation\[[MP69](https://arxiv.org/html/2607.05283#bib.bib7)\]\. Moody later proved thatρn\\rho\_\{n\}is not faithful forn≥9n\\geq 9\[[MOO91](https://arxiv.org/html/2607.05283#bib.bib8)\]\. Long–Paton built on Moody’s ideas to improve this ton≥6n\\geq 6\[[LP93](https://arxiv.org/html/2607.05283#bib.bib4)\], and Bigelow later added the casen=5n=5\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10)\]\.

In the years following Bigelow’s paper there have been many attempts to detect elements in the kernel of the remaining caseρ4\\rho\_\{4\}, or to narrow the search for such, each of which contributed to our understanding of this question as viewed through the lens of various areas of mathematics; see for example Alperin–Farb–Noskov\[[AFN02](https://arxiv.org/html/2607.05283#bib.bib79)\], Beridze–Traczyk\[[BT18a](https://arxiv.org/html/2607.05283#bib.bib34),[BT18b](https://arxiv.org/html/2607.05283#bib.bib37)\], Calvez–Ito\[[CI17](https://arxiv.org/html/2607.05283#bib.bib36)\], Datta\[[DAT22](https://arxiv.org/html/2607.05283#bib.bib38)\], Dlugie\[[DLU24](https://arxiv.org/html/2607.05283#bib.bib14)\], Fullarton–Shadrach\[[FS19](https://arxiv.org/html/2607.05283#bib.bib56)\], Gibson–Williamson–Yacobi\[[GWY25](https://arxiv.org/html/2607.05283#bib.bib15)\], and Witzel–Zaremsky\[[WZ15](https://arxiv.org/html/2607.05283#bib.bib35)\]\. The question of faithfulness forρ4\\rho\_\{4\}also appears as Question 3\.1 in Margalit’s problem list for mapping class groups\[[MAR19](https://arxiv.org/html/2607.05283#bib.bib33)\]\.

###### Main Theorem\.

The Burau representationρ4\\rho\_\{4\}ofB4\\operatorname\{B\}\_\{4\}is faithful\.

As a first step to proving this result, we will give a new and simple topological proof of the faithfulness of the Burau representation in the casen=3n=3\. We will then adapt these ideas to the case ofn=4n=4by exploiting the structure of point\-pushing maps inB4\\operatorname\{B\}\_\{4\}\.

This approach is in contrast to earlier work onBn\\operatorname\{B\}\_\{n\}forn≥5n\\geq 5and attempts to prove that the Burau representationρ4\\rho\_\{4\}is not faithful\. In particular, we do not consider elements in the image ofρ4\\rho\_\{4\}that potentially generate a free group\. We refer the reader to the second author’s book\[[BIR74](https://arxiv.org/html/2607.05283#bib.bib86), Theorem 3\.19\], where this approach was first introduced\. In addition to the papers mentioned above, see also Bigelow\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10), Section 3\]and Moran\[[MOR91](https://arxiv.org/html/2607.05283#bib.bib71)\]for details of other approaches\.

#### Application to the Jones representation\.

In his seminal work, Jones introduced a representation of the braid group that contains the \(reduced\) Burau representation as a summand; for a full definition we refer the reader to his paper\[[JON87](https://arxiv.org/html/2607.05283#bib.bib78)\]\. As Jones points out, the faithfulness of Burau representationρn\\rho\_\{n\}for anynnimplies the faithfulness of the Jones representation of the braid groupBnB\_\{n\}\. Therefore we immediately obtain the following corollary of our Main Theorem\.

###### Corollary 1\.1\.

The Jones representation ofBn\\operatorname\{B\}\_\{n\}is faithful forn=4n=4\.

We also refer the reader to Kasahara’s subsequent work on the Jones representation, in particular to his explanation of the equivalence between the faithfulness of the Burau representation and the Jones representation in the casen=4n=4\[[KAS08](https://arxiv.org/html/2607.05283#bib.bib2), Remark 5\.6\]\.

#### Strategy of proof\.

LetKiK\_\{i\}denote the point\-pushing subgroup ofBn\\operatorname\{B\}\_\{n\}obtained as the kernel of the Birman exact sequence for the diskDnD\_\{n\}by “forgetting” theit​hi^\{th\}marked point in the diskDnD\_\{n\}; see Section[5](https://arxiv.org/html/2607.05283#S5)for more details\. The intersection∩i=1nKi\\cap\_\{i=1\}^\{n\}K\_\{i\}of all point\-pushing subgroups inBn\\operatorname\{B\}\_\{n\}is known as theBrunnian groupBrunn\\operatorname\{Brun\}\_\{n\}, and it is a normal subgroup ofBn\\operatorname\{B\}\_\{n\}\. A theorem of Long states that the Burau representionρn\\rho\_\{n\}is faithful onBnB\_\{n\}if it is faithful on any nontrivial noncentral normal subgroup ofBn\\operatorname\{B\}\_\{n\}\[[LON86](https://arxiv.org/html/2607.05283#bib.bib3), Theorem 2\.2\]\. Suppose now thatker⁡\(ρ4\)\\ker\(\\rho\_\{4\}\)were nontrivial\. By Long’s theorem,ker⁡\(ρ4\)\\ker\(\\rho\_\{4\}\)intersectsBrun4\\operatorname\{Brun\}\_\{4\}nontrivially\.

###### Proposition 1\.2\.

If the Burau representationρ4\\rho\_\{4\}is faithful on its restriction to the Brunnian groupBrun4\\operatorname\{Brun\}\_\{4\}, then it is faithful onB4\\operatorname\{B\}\_\{4\}\.

Building on ideas of Long–Paton and Bigelow, we associate to each braid inBn\\operatorname\{B\}\_\{n\}a sequence of disks inDnD\_\{n\}that carry certain combinatorial data\. We use this technology to give a new proof of Magnus–Peluso’s theorem that the Burau representationρ3\\rho\_\{3\}is faithful\[[MP69](https://arxiv.org/html/2607.05283#bib.bib7)\]\. Our proof yields the more general result that anynn\-strand braid whose associated disk sequence satisfies a certainparity condition, defined in Section[4](https://arxiv.org/html/2607.05283#S4), does not lie in the kernel ofρn\\rho\_\{n\}\.

Braids inBn\\operatorname\{B\}\_\{n\}do not in general satisfy this parity condition whenn≥4n\\geq 4\. However, we show that whenn=4n=4, the parity condition “almost” holds for any point\-pushing braidΦ∈K4\\Phi\\in K\_\{4\}that admits a factorization as aproper productof certain push\-maps\. The next step is to “correct”Φ\\Phiby embeddingK4K\_\{4\}inB5\\operatorname\{B\}\_\{5\}, that is, we useΦ\\Phito construct a 5\-braid satisfying the parity condition\. We then apply a result of Moody to conclude that the original braidΦ∈K4\\Phi\\in K\_\{4\}does not lie in the kernel ofρ4\\rho\_\{4\}\. The final step is to show that any braid inK4K\_\{4\}is conjugate to another braid inK4K\_\{4\}that can be realized as a proper product of push\-maps\.

Finally, we remark that,a priori, it is not necessarily clear thatBrunn\\operatorname\{Brun\}\_\{n\}is nontrivial\. The simplest example of a Brunnian braid is a three\-strand braid that closes to form the Borromean rings\. Figure[1\.1](https://arxiv.org/html/2607.05283#S1.F1)gives an example of a nontrivial element ofBrun4\\operatorname\{Brun\}\_\{4\}, the case of interest to us, and a similar construction yields nontrivial examples of Brunnian braids inBn\\operatorname\{B\}\_\{n\}for anynn\. Moreover, Whittlesey has shown that all nontrivial Brunnian braids are pseudo\-Anosov\[[WHI00](https://arxiv.org/html/2607.05283#bib.bib77)\]\. We note that Whittlesey’s results are stated for the mapping class group of the\(n\+1\)\(n\+1\)\-punctured sphere, but the result holds inBn\\operatorname\{B\}\_\{n\}; see Lee\-Song’s discussion of this point\[[LS05](https://arxiv.org/html/2607.05283#bib.bib57), Section 1\]\.

![Refer to caption](https://arxiv.org/html/2607.05283v1/brunnian-braid.png)Figure 1\.1:A Brunnian 4\-braidβ\\beta\.
#### Kernel and image of Burau\.

There is a great deal of literature on the faithfulness of the Burau representation, and this paper settles that question in the final outstanding case\. The questions of giving both a useful characterization of its kernel and of its image remain wide open in general, and the authors believe that the methods of this paper may be useful in achieving progress on these important questions\.

#### Outline of the paper\.

In Section[2](https://arxiv.org/html/2607.05283#S2), we introduce the Moody polynomial of a braid, an invariant that provides an obstruction for an element to lie in the kernel of the Burau representation\. Building on Moody’s ideas and their development by Long–Paton, Bigelow later introduced combinatorial tools that are useful for computing the obstruction; we describe these in Section[3](https://arxiv.org/html/2607.05283#S3)\. Using these tools, in Section[4](https://arxiv.org/html/2607.05283#S4), we give our new topological proof of faithfulness of the Burau representation in the casen=3n=3\. We introduce our key technical tool,proper productsof point\-pushing braids, in Section[5](https://arxiv.org/html/2607.05283#S5)\. In Section[6](https://arxiv.org/html/2607.05283#S6)we construct point\-pushing maps arising as particular proper products, and use these to establish that the Burau representationρ4\\rho\_\{4\}is faithful on its restriction to the point\-pushing groupBrun4\\operatorname\{Brun\}\_\{4\}; by Proposition[1\.2](https://arxiv.org/html/2607.05283#S1.Thmtheorem2), this completes the proof of the Main Theorem\. Finally, we give an example illustrating the constructions used in our proof in Section[7](https://arxiv.org/html/2607.05283#S7)\.

Acknowledgements\.The authors heartily thank Dan Margalit for many helpful discussions and are especially grateful for his extensive and valuable feedback on initial drafts of this paper\. The second author would also like to thank him for his interest in this project from its inception seven years ago, when the first two authors had begun to work on it together\. The authors also thank Michael Dougherty and Benson Farb for additional helpful comments on an early draft and Ken Birman for further helpful remarks\. The first author thanks David Gabai for his invaluable mentorship and constant encouragement\. The third author is also grateful to the Institute for Computational and Experimental Research in Mathematics \(ICERM\) at Brown University, where in 2022 she was a participant in the semester\-long program “Braids”, during which initial discussions that led to her joining this project took place, and to Muffy Calder for her support and encouragement throughout\. AI was used to proofread a draft of the manuscript\.

## 2The Moody polynomial and winding numbers

While Burau originally defined his representation by giving its values on the standard Artin generators ofBn\\operatorname\{B\}\_\{n\}\(see, for example,\[[BB05](https://arxiv.org/html/2607.05283#bib.bib18), Section 4\.2\]\), it will be more useful for present purposes to defineρn\\rho\_\{n\}via the action ofBn\\operatorname\{B\}\_\{n\}on the diskDnD\_\{n\}, which we consider here as annn\-times punctured disk\. Letp∗p\_\{\*\}denote a point in the boundary of the diskDnD\_\{n\}, and letx1,…,xnx\_\{1\},\\dots,x\_\{n\}denote the standard free generators ofπ1​\(Dn,p∗\)\\pi\_\{1\}\(D\_\{n\},p\_\{\*\}\)\. We consider the mappingπ1​\(Dn,p∗\)→ℤ\\pi\_\{1\}\(D\_\{n\},p\_\{\*\}\)\\to\\mathbb\{Z\}that takes a word inx1,…​xnx\_\{1\},\\dots x\_\{n\}to the sum of its exponents, and letDn~\\widetilde\{D\_\{n\}\}denote the covering space associated with its kernel\.

The group of covering transformations ofDn~\\widetilde\{D\_\{n\}\}is isomorphic toℤ\\mathbb\{Z\}, which we denote as a multiplicative group generated bytt\. Following the treatment of Long–Paton, it will be convenient for us to work with the relative homology groupH1​\(Dn~,\{p∗~\}\)H\_\{1\}\(\\widetilde\{D\_\{n\}\},\\\{\\widetilde\{p\_\{\*\}\}\\\}\), where\{p∗~\}\\\{\\widetilde\{p\_\{\*\}\}\\\}denotes the full pre\-image of the basepointp∗p\_\{\*\}in the coverDn~\\widetilde\{D\_\{n\}\}\. The braid groupBn\\operatorname\{B\}\_\{n\}acts on theℤ​\[t,t−1\]\\mathbb\{Z\}\[t,t^\{\-1\}\]\-moduleH1​\(Dn~,\{p∗~\}\)H\_\{1\}\(\\widetilde\{D\_\{n\}\},\\\{\\widetilde\{p\_\{\*\}\}\\\}\), which is free of ranknn; this action is theunreduced Burau representationρn\\rho\_\{n\}, which we will refer to simply as the Burau representation\.

We next record a basic fact that follows directly from the definition ofρn\\rho\_\{n\}\.

###### Observation 2\.1\.

Letf:Bn→Bn\+1f:\\operatorname\{B\}\_\{n\}\\rightarrow\\operatorname\{B\}\_\{n\+1\}be the inclusion map corresponding to the standard embedding ofDnD\_\{n\}intoDn\+1D\_\{n\+1\}\. IfΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}lies in the kernel ofρn\\rho\_\{n\}, thenf​\(Φ\)f\(\\Phi\)lies in the kernel ofρn\+1\\rho\_\{n\+1\}\.

Observation[2\.1](https://arxiv.org/html/2607.05283#S2.Thmtheorem1)will be crucial in our proof of the Main Theorem\.

\\begin\{overpic\}\[width=346\.89731pt\]\{covering\-space\-landscape\.png\} \\put\(\-7\.0,14\.0\)\{$\\dots$\} \\par\\put\(1\.5,15\.5\)\{\\tiny$\+$\} \\put\(4\.0,15\.5\)\{\\tiny$\-$\} \\put\(8\.0,15\.5\)\{\\tiny$\+$\} \\put\(10\.7,15\.5\)\{\\tiny$\-$\} \\put\(14\.8,15\.5\)\{\\tiny$\+$\} \\put\(17\.7,15\.5\)\{\\tiny$\-$\} \\put\(21\.0,15\.5\)\{\\tiny$\+$\} \\put\(23\.2,15\.5\)\{\\tiny$\-$\} \\par\\put\(39\.0,15\.5\)\{\\tiny$\+$\} \\put\(42\.0,15\.5\)\{\\tiny$\-$\} \\put\(44\.9,15\.5\)\{\\tiny$\+$\} \\put\(48\.0,15\.5\)\{\\tiny$\-$\} \\put\(52\.0,15\.5\)\{\\tiny$\+$\} \\put\(55\.0,15\.5\)\{\\tiny$\-$\} \\put\(58\.0,15\.5\)\{\\tiny$\+$\} \\put\(61\.0,15\.5\)\{\\tiny$\-$\} \\par\\put\(74\.5,15\.5\)\{\\tiny$\+$\} \\put\(77\.0,15\.5\)\{\\tiny$\-$\} \\put\(80\.9,15\.5\)\{\\tiny$\+$\} \\put\(83\.5,15\.5\)\{\\tiny$\-$\} \\put\(88\.0,15\.5\)\{\\tiny$\+$\} \\put\(91\.0,15\.5\)\{\\tiny$\-$\} \\put\(94\.0,15\.5\)\{\\tiny$\+$\} \\put\(96\.5,15\.5\)\{\\tiny$\-$\} \\par\\put\(12\.0,\-3\.0\)\{$t^\{0\}$\} \\put\(49\.0,\-3\.0\)\{$t^\{1\}$\} \\put\(86\.0,\-3\.0\)\{$t^\{2\}$\} \\par\\put\(103\.0,14\.0\)\{$\\dots$\} \\par\\put\(3\.0,10\.0\)\{$p\_\{1\}$\} \\put\(9\.0,10\.0\)\{$p\_\{2\}$\} \\put\(16\.0,10\.0\)\{$p\_\{3\}$\} \\put\(22\.0,10\.0\)\{$p\_\{4\}$\} \\par\\put\(40\.0,10\.0\)\{$p\_\{1\}$\} \\put\(47\.0,10\.0\)\{$p\_\{2\}$\} \\put\(53\.0,10\.0\)\{$p\_\{3\}$\} \\put\(59\.0,10\.0\)\{$p\_\{4\}$\} \\par\\put\(76\.0,10\.0\)\{$p\_\{1\}$\} \\put\(82\.0,10\.0\)\{$p\_\{2\}$\} \\put\(88\.0,10\.0\)\{$p\_\{3\}$\} \\put\(94\.0,10\.0\)\{$p\_\{4\}$\} \\par\\end\{overpic\}Figure 2\.1:The zeroth, first, and second “decks” in the universal cyclic coverD4~\\widetilde\{D\_\{4\}\}of the diskD4D\_\{4\}\.Moody polynomials\.In what follows, we will closely follow Bigelow’s notation and exposition\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10)\]\. We will use the termarcto refer to a proper embedding of an interval in the diskDnD\_\{n\}andsubarcto refer to its restriction to a compact connected subinterval\. Our arcs will normally be oriented, and it will often be convenient to consider arcs in terms of their image in the diskDnD\_\{n\}rather than as functions per se\.

Letα\\alphaandβ\\betabe two oriented arcs inDnD\_\{n\}, considered as a disk with marked points, whose endpoints lie in the set\{p1,…,pn,p∗\}\\\{p\_\{1\},\\ldots,p\_\{n\},p\_\{\*\}\\\}, and choose two corresponding liftsα~\\tilde\{\\alpha\}andβ~\\tilde\{\\beta\}inDn~\\widetilde\{D\_\{n\}\}\. We define theMoody polynomial𝕄​\(α,β\)∈ℤ​\[t,t−1\]\\mathbb\{M\}\(\\alpha,\\beta\)\\in\\mathbb\{Z\}\[t,t^\{\-1\}\]of the oriented arcsα\\alphaandβ\\betaas follows:

𝕄​\(α,β\)=∑ℓ∈ℤ\(tℓ⋅α~,β~\)​tℓ\\mathbb\{M\}\(\\alpha,\\beta\)=\\sum\_\{\\ell\\in\\mathbb\{Z\}\}\(t^\{\\ell\}\\cdot\\tilde\{\\alpha\},\\tilde\{\\beta\}\)t^\{\\ell\}where\(tℓ⋅α~,β~\)\(t^\{\\ell\}\\cdot\\tilde\{\\alpha\},\\tilde\{\\beta\}\)denotes the algebraic intersection number of the lifttℓ⋅α~t^\{\\ell\}\\cdot\\tilde\{\\alpha\}withβ~\\tilde\{\\beta\}in the coverDn~\\widetilde\{D\_\{n\}\}\. We note that the algebraic intersection number of arcs does not include intersections at any shared endpoints; in our case, we will always choose our arcs so that the endpoints ofα\\alphaare disjoint from those ofβ\\beta\. We also note that the Moody polynomial𝕄​\(α,β\)\\mathbb\{M\}\(\\alpha,\\beta\)is only well defined up to multiplication by a power oftt, due to its dependence on our choice of the liftsα~\\tilde\{\\alpha\}andβ~\\tilde\{\\beta\}\. Our convention will be to choose our lifts so that the first point at whichβ\\betacrossesα\\alphais assigned the monomial±t0\\pm t^\{0\}\.

Following Bigelow\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10)\]and Long–Paton\[[LP93](https://arxiv.org/html/2607.05283#bib.bib4)\], we now describe a planar interpretation of the Moody polynomial that enables us to work entirely in the base spaceDnD\_\{n\}rather then in the cover\. Without loss of generality, we can assume that the oriented arcsα\\alphaandβ\\betaintersect transversely in finitely many points; we label theseq1,…,qmq\_\{1\},\\ldots,q\_\{m\}according to the order in which they appear as we traverse the arcβ\\betaaccording to its orientation\. Each intersection pointqiq\_\{i\}corresponds to a single point of intersection between two liftsα~\\tilde\{\\alpha\}andβ~\\tilde\{\\beta\}, and hence each pointqiq\_\{i\}contributes a monomial±tki\\pm t^\{k\_\{i\}\}to the overall sum in the Moody polynomial\. Here the exponentkik\_\{i\}is such thatβ~\\tilde\{\\beta\}andtki⋅α~t^\{k\_\{i\}\}\\cdot\\tilde\{\\alpha\}cross at a lift ofqiq\_\{i\}, and the signϵi\\epsilon\_\{i\}of the monomial is the sign of that crossing whereϵi∈\{−1,1\}\\epsilon\_\{i\}\\in\\\{\-1,1\\\}\. Then an equivalent formulation of the Moody polynomial of the arcsα\\alphaandβ\\betais as follows:

𝕄​\(α,β\)=∑i=1mϵi​tki\.\\mathbb\{M\}\(\\alpha,\\beta\)=\\sum\_\{i=1\}^\{m\}\\epsilon\_\{i\}t^\{k\_\{i\}\}\.We emphasize that it may happen thatki=kjk\_\{i\}=k\_\{j\}fori≠ji\\neq j\. Indeed, our business in this paper will be to determine conditions under which this could happen\. When we write the Moody polynomial as the sum ofmmterms of the formϵi​tki\\epsilon\_\{i\}t^\{k\_\{i\}\}without combining like terms, we will refer to this as theunsimplified Moody polynomial; when we combine all like terms we will refer to this as thesimplified Moody polynomial\. Moody showed that the polynomial𝕄​\(α,β\)\\mathbb\{M\}\(\\alpha,\\beta\)completely encodes the faithfulness of the Burau representation\[[MOO91](https://arxiv.org/html/2607.05283#bib.bib8)\]\. We will use one direction of his characterization; see Theorem[2\.3](https://arxiv.org/html/2607.05283#S2.Thmtheorem3)\.

Conventions and color\-coding\.For the remainder of this paper, we will always choose the arcα\\alphato be the horizontal arc joiningp1p\_\{1\}top2p\_\{2\}oriented from left to right, and we will color it blue in all figures, as shown in Figure[2\.2](https://arxiv.org/html/2607.05283#S2.F2)\. We also letβ∗3\\beta\_\{\*\}^\{3\}denote the oriented arc shown in Figure[2\.2](https://arxiv.org/html/2607.05283#S2.F2)from the basepointp∗p\_\{\*\}to the puncturep3p\_\{3\}and color it red\.

\\begin\{overpic\}\[scale=\{1\.7\}\]\{disk\-alpha\-beta\.png\} \\put\(22\.0,40\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(50\.0,20\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta\_\{\*\}^\{3\}$\}\} \\put\(38\.0,\-5\.0\)\{$p\_\{\*\}$\} \\put\(9\.0,53\.0\)\{$p\_\{1\}$\} \\put\(35\.0,53\.0\)\{$p\_\{2\}$\} \\put\(60\.0,53\.0\)\{$p\_\{3\}$\} \\put\(85\.0,53\.0\)\{$p\_\{4\}$\} \\end\{overpic\}Figure 2\.2:The blue arcα\\alphajoinsp1p\_\{1\}top2p\_\{2\}\. The red arcβ∗3\\beta\_\{\*\}^\{3\}joins the basepoint on∂D\\partial Dto the pointp3p\_\{3\}\.When we apply a braidΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}to the arcβ∗3\\beta\_\{\*\}^\{3\}, we will usually denote this byβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phiand also colorβ\\betared; note that we writeΦ\\Phion the right of the arc to which we are applying it because we will follow the tradition of writing multiplication of braids from left to right\.

Furthermore, from this point on, a blue \(respectively red\) arc will always be a subarc ofα\\alpha\(respectivelyβ\\beta\)\. Later we will add gold to our list of special colors\. Our sign convention will be to say thatqiq\_\{i\}is a positive crossing, that is,ϵi=1\\epsilon\_\{i\}=1, ifβ\\betais directed downwards atqiq\_\{i\}and thatqiq\_\{i\}is a negative crossing withϵi=−1\\epsilon\_\{i\}=\-1ifβ\\betais directed upwards atqiq\_\{i\}\.

For any arcsγ\\gammaandδ\\deltain the diskDnD\_\{n\}, we letι​\(γ,δ\)\\iota\(\\gamma,\\delta\)denote their geometric intersection number\. Consider now our fixed choice of arcα\\alpha\. A remarkable result of Moody states that the non\-faithfulness of the Burau representation is equivalent to the existence of an arcβ\\betajoining the basepointp∗p\_\{\*\}to the marked pointp3p\_\{3\}inDnD\_\{n\}such thatι​\(α,β\)\>0\\iota\(\\alpha,\\beta\)\>0and𝕄​\(α,β\)=0\\mathbb\{M\}\(\\alpha,\\beta\)=0\[[MOO91](https://arxiv.org/html/2607.05283#bib.bib8)\]\. With this in mind, we define theMoody polynomial of the braidΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}as follows:

𝕄Φ=𝕄​\(α,\(β∗3\)​Φ\)\.\\displaystyle\\mathbb\{M\}\_\{\\Phi\}=\\mathbb\{M\}\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\Phi\)\.We remark that by taking the boundary of a regular neighborhood of the arc\(β∗3\)​Φ\(\\beta\_\{\*\}^\{3\}\)\\Phi, we can view as a loop based atp∗p\_\{\*\}traveling aroundp3p\_\{3\}, and that𝕄Φ\\mathbb\{M\}\_\{\\Phi\}is well defined on the homology class of a lift of\[\(β∗3\)​Φ\]\[\(\\beta\_\{\*\}^\{3\}\)\\Phi\]inH1​\(Dn~,\{p∗~\}\)H\_\{1\}\(\\widetilde\{D\_\{n\}\},\\\{\\widetilde\{p\_\{\*\}\}\\\}\), which we denote by\[\(β∗3\)​Φ\]\[\(\\beta\_\{\*\}^\{3\}\)\\Phi\]\. We also emphasize that an elementΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}uniquely determines an arcβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phi, but the converse does not hold: there exist infinitely many pairsΦ1,Φ2∈Bn\\Phi\_\{1\},\\Phi\_\{2\}\\in\\operatorname\{B\}\_\{n\}for which\(β∗3\)​Φ1=\(β∗3\)​Φ2\(\\beta\_\{\*\}^\{3\}\)\\Phi\_\{1\}=\(\\beta\_\{\*\}^\{3\}\)\\Phi\_\{2\}\. We refer the reader to Long–Paton\[[LP93](https://arxiv.org/html/2607.05283#bib.bib4), Section 1\]and Bigelow\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10), Theorem 1\.4\]for further details\. For our purposes, we do not need the full strength of Moody’s result; we will only require one direction of the equivalence\.

###### Theorem 2\.2\(Moody\)\.

Letn≥3n\\geq 3\. If, given any oriented arcβ\\betafromp∗p\_\{\*\}top3p\_\{3\}such thatι​\(α,β\)\>0\\iota\(\\alpha,\\beta\)\>0, we have that𝕄​\(α,β\)≠0\\mathbb\{M\}\(\\alpha,\\beta\)\\neq 0, then the Burau representationρn\\rho\_\{n\}is faithful\. Equivalently, if we have that𝕄Φ≠0\\mathbb\{M\}\_\{\\Phi\}\\neq 0, for any braidΦ\\Phisuch thatΦ​\(p3\)=p3\\Phi\(p\_\{3\}\)=p\_\{3\}andι​\(α,\(β3∗\)​Φ\)\>0\\iota\(\\alpha,\(\\beta\_\{3\}^\{\*\}\)\\Phi\)\>0, then the Burau representationρn\\rho\_\{n\}is faithful\.

The equivalence of the two statements follows from the fact that every oriented arcβ\\betafromp∗p\_\{\*\}top3p\_\{3\}arises asβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phifor someΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}by the change of coordinates principle; note that such a braidΦ\\Phinecessarily fixes the pointp3p\_\{3\}\. It will be useful for our purposes to adapt Moody’s theorem to obtain a specific obstruction to an individual braid lying in the kernel ofρn\\rho\_\{n\}, for which we give a short proof for the sake of completeness\.

###### Theorem 2\.3\(Moody\)\.

LetΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}forn≥3n\\geq 3\. If𝕄Φ⋅Γ≠𝕄Γ\\mathbb\{M\}\_\{\\Phi\\cdot\\Gamma\}\\neq\\mathbb\{M\}\_\{\\Gamma\}for someΓ∈Bn\\Gamma\\in B\_\{n\}, thenΦ\\Phidoes not lie in the kernel ofρn\\rho\_\{n\}\.

###### Proof\.

Suppose thatΦ\\Philies in the kernel ofρn\\rho\_\{n\}\. ThenΓ−1⋅Φ⋅Γ\\Gamma^\{\-1\}\\cdot\\Phi\\cdot\\Gammamust also lie in the kernel\. We have that\(β∗3\)⋅Γ\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gammacorresponds to an element\[\(β∗3\)⋅Γ\]\[\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma\]in the homology of the coverDn~\\widetilde\{D\_\{n\}\}relative to the pre\-image of a basepoint\. SinceΓ−1⋅Φ⋅Γ\\Gamma^\{\-1\}\\cdot\\Phi\\cdot\\Gammaacts trivially on this homology group, we have that\[\(β∗3\)⋅\(Φ⋅Γ\)\]=\[\(\(β∗3\)⋅Γ\)⋅\(Γ−1⋅Φ⋅Γ\)\]=\[\(β∗3\)⋅Γ\]\[\(\\beta\_\{\*\}^\{3\}\)\\cdot\(\\Phi\\cdot\\Gamma\)\]=\[\(\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma\)\\cdot\(\\Gamma^\{\-1\}\\cdot\\Phi\\cdot\\Gamma\)\]=\[\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma\]\. The result now follows from our previous observation that𝕄Φ\\mathbb\{M\}\_\{\\Phi\}well defined on the homology class\[\(β∗3\)​Φ\]\[\(\\beta\_\{\*\}^\{3\}\)\\Phi\]inH1​\(Dn~,\{p∗~\}\)H\_\{1\}\(\\widetilde\{D\_\{n\}\},\\\{\\widetilde\{p\_\{\*\}\}\\\}\)\. ∎

In Section[6](https://arxiv.org/html/2607.05283#S6)we will identify certain products of braids inB4\\operatorname\{B\}\_\{4\}to which we will then apply Theorem[2\.3](https://arxiv.org/html/2607.05283#S2.Thmtheorem3)in order to establish the faithfulness ofρ4\\rho\_\{4\}\. Moreover, as discussed in Section[1](https://arxiv.org/html/2607.05283#S1), we ultimately reduce our search to the Brunnian subgroupBrunn\\operatorname\{Brun\}\_\{n\}inBn\\operatorname\{B\}\_\{n\}, where all nontrivial braids are pseudo\-Anosov, and hence for allΦ∈Brunn\\Phi\\in\\operatorname\{Brun\}\_\{n\}we have thatι​\(α,\(β∗3\)⋅Φ\)\>0\\iota\(\\alpha,\(\\beta^\{3\}\_\{\*\}\)\\cdot\\Phi\)\>0\. We will also use the following special case of Theorem[2\.3](https://arxiv.org/html/2607.05283#S2.Thmtheorem3)\.

###### Corollary 2\.4\.

LetΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}forn≥3n\\geq 3\. If𝕄Φ≠0\\mathbb\{M\}\_\{\\Phi\}\\neq 0, thenΦ\\Phidoes not lie in the kernel ofρn\\rho\_\{n\}\.

## 3Disk sequences and total winding number

Letβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phi, and as above, we label the points of intersection ofα\\alphawithβ\\betabyq1,…,qmq\_\{1\},\\ldots,q\_\{m\}according to the order in which we encounter them as we travel alongβ\\beta\. Fori=1,…​m−1i=1,\\ldots m\-1, we letαi\\alpha\_\{i\}andβi\\beta\_\{i\}denote the subarcs ofα\\alphaandβ\\beta, respectively, that join the intersection pointqiq\_\{i\}toqi\+1q\_\{i\+1\}\. Note that, with our choice of notation, this means that each arcβi\\beta\_\{i\}contains no intersection pointqjq\_\{j\}in its interior, while an arcαi\\alpha\_\{i\}may contain a number of other intersection points in its interior\.

By construction, for eachi=1,…​m−1i=1,\\ldots m\-1, we have thatαi∪βi\\alpha\_\{i\}\\cup\\beta\_\{i\}bounds akk\-punctured diskΔi\\Delta\_\{i\}\. We define thetotal winding numberWiW\_\{i\}associated to the diskΔi\\Delta\_\{i\}to be equal tokkifβi\\beta\_\{i\}is oriented clockwise with respect toΔi\\Delta\_\{i\}, or to be equal to−k\-kotherwise\. We will refer to the sequenceW1,…,Wm−1W\_\{1\},\\ldots,W\_\{m\-1\}as thewinding number sequenceofΦ\\Phi, and to the sequenceΔ1,Δ2,…,Δm−1\\Delta\_\{1\},\\Delta\_\{2\},\\ldots,\\Delta\_\{m\-1\}as thedisk sequenceofΦ\\Phi; see Figure[3\.1](https://arxiv.org/html/2607.05283#S3.F1)for examples\.

We will consider two such disksΔi,Δj\\Delta\_\{i\},\\Delta\_\{j\}to beequivalentif there is an isotopy of\(Dn,α\)\(D\_\{n\},\\alpha\)taking theβ\\beta\-component of∂Δi\\partial\\Delta\_\{i\}to the theβ\\beta\-component of∂Δj\\partial\\Delta\_\{j\}; in other wordsα\\alphais fixed setwise and the endpoints of theβ\\beta\-arc must be contained inα\\alphaat each level of the isotopy\)\. We note that this notion of equivalence corresponds to allowing isotopies of the arcβ\\betain the diskDnD\_\{n\}\.

We will see that the combinatorial data of the winding number sequence is sufficient for determining the faithfulness of the Burau representation in the casen=3n=3, and that the more detailed information of the disk sequence is required in the casen=4n=4\. The following lemma appears as a remark in Bigelow’s paper\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10), Section 3\]\. His remark describes the relationship between winding numbers and the degreekik\_\{i\}of the monomial that each intersection pointqiq\_\{i\}contributes to the Moody polynomial\.

![Refer to caption](https://arxiv.org/html/2607.05283v1/cp-example-1.png)\(a\)
![Refer to caption](https://arxiv.org/html/2607.05283v1/cp-example-2.png)\(b\)

Figure 3\.1:The arcsα\\alphaandβ\\betain \(a\) give rise to a singleton winding number sequence given byW1=1W\_\{1\}=1, with corresponding Moody polynomial1−t1\-t\. The winding number sequence for the arcs in \(b\) is given by1,1,−1,−2,11,1,\-1,\-2,1; its Moody polynomial is2−2​t\+t2−t−12\-2t\+t^\{2\}\-t^\{\-1\}\.###### Lemma 3\.1\(Bigelow\[[BIG99](https://arxiv.org/html/2607.05283#bib.bib10)\]\)\.

Letqiq\_\{i\}andqi\+1q\_\{i\+1\}be two points inα∩β\\alpha\\cap\\betathat are consecutive with respect to the oriented arcβ\\beta, and letkik\_\{i\}andki\+1k\_\{i\+1\}be the exponents of the monomials corresponding toqiq\_\{i\}andqi\+1q\_\{i\+1\}\. IfWiW\_\{i\}denotes theit​hi^\{th\}term in the winding number sequence forβ\\beta, then we have:

Wi=ki\+1−ki\.W\_\{i\}=k\_\{i\+1\}\-k\_\{i\}\.\(1\)

We record here the following useful corollary of Lemma[3\.1](https://arxiv.org/html/2607.05283#S3.Thmtheorem1), also observed by Bigelow:

###### Corollary 3\.2\.

LetW1,…,WmW\_\{1\},\\ldots,W\_\{m\}be a winding number sequence for a braid inBnB\_\{n\}, and letkik\_\{i\}andkjk\_\{j\}be the exponents of the Moody monomials corresponding to two pointsqiq\_\{i\}andqjq\_\{j\}inα∩β\\alpha\\cap\\beta\. Thenki=kjk\_\{i\}=k\_\{j\}if and only if∑k=ij−1Wk=0\\sum\_\{k=i\}^\{j\-1\}W\_\{k\}=0\.

Equipped with this useful criterion for tracking repeats of an exponent occurring in the Moody polynomial, we will proceed in the next section to consider the question of faithfulness ofρn\\rho\_\{n\}whenn=3n=3\.

## 4Faithfulness for three strands

The following theorem was first proved by Magnus and Peluso in 1969 using purely algebraic methods\. We give a new proof\.

###### Theorem 4\.1\(Magnus–Peluso\[[MP69](https://arxiv.org/html/2607.05283#bib.bib7)\]\)\.

The Burau representationρ3\\rho\_\{3\}is faithful\.

###### Proof\.

Using the same notation as in the previous section, we letΦ∈B3\\Phi\\in\\operatorname\{B\}\_\{3\}be an element such that the geometric intersection ofα\\alphaandβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phiis equal tom\>0m\>0\. By Theorem[2\.2](https://arxiv.org/html/2607.05283#S2.Thmtheorem2), it suffices to show that any single term of the simplified Moody polynomial𝕄Φ=∑i=1mϵi​tki\\mathbb\{M\}\_\{\\Phi\}=\\sum\_\{i=1\}^\{m\}\\epsilon\_\{i\}t^\{k\_\{i\}\}is nonzero\. Ifm=1m=1, then the Moody polynomial consists of a single linear term, and hence is nonzero\. Suppose now thatm\>1m\>1, and letW1,…,Wm−1W\_\{1\},\\ldots,W\_\{m\-1\}denote the winding number sequence of the braidΦ\\Phi\. Each corresponding disk is bounded by a subarc ofα\\alphaand a subarc ofβ\\betaand contains 1, 2, or 3 marked points\. The assumption thatα\\alphaandβ\\betaare in minimal position then implies that, up to reflection about the coordinate axes and up to homeomorphism of the disk preserving each marked pointpip\_\{i\}pointwise and the arcα\\alphasetwise, each disk in the disk sequence is equivalent to one of the three types of regions illustrated in Figure[4\.1](https://arxiv.org/html/2607.05283#S4.F1), possibly after applying a symmetry\.

![Refer to caption](https://arxiv.org/html/2607.05283v1/cp-1-pun.png)\(a\)\|Wi\|=1\|W\_\{i\}\|=1
![Refer to caption](https://arxiv.org/html/2607.05283v1/cp-2-pun.png)\(b\)\|Wi\|=2\|W\_\{i\}\|=2
![Refer to caption](https://arxiv.org/html/2607.05283v1/cp-3-pun.png)\(c\)\|Wi\|=3\|W\_\{i\}\|=3

Figure 4\.1:Possible disks in the disk sequence of an element ofB3\\operatorname\{B\}\_\{3\}\.Notice that for eachiisuch that\|Wi\|=2\|W\_\{i\}\|=2, the signϵi\\epsilon\_\{i\}of the two points of intersection ofβ\\betawithα\\alphamust be the same, that is,ϵi=ϵi\+1\\epsilon\_\{i\}=\\epsilon\_\{i\+1\}\. Furthermore, if\|Wi\|∈\{1,3\}\|W\_\{i\}\|\\in\\\{1,3\\\}, then we have thatϵi=−ϵi\+1\\epsilon\_\{i\}=\-\\epsilon\_\{i\+1\}\.

Suppose now thatki=kjk\_\{i\}=k\_\{j\}for two distinct terms in the unsimplified Moody polynomial\. Then∑ℓ=ij−1Wℓ=0\\sum\_\{\\ell=i\}^\{j\-1\}W\_\{\\ell\}=0by Corollary[3\.2](https://arxiv.org/html/2607.05283#S3.Thmtheorem2)\. Hence fori≤ℓ≤j−1i\\leq\\ell\\leq j\-1we must have an even number of termsWℓW\_\{\\ell\}with\|Wℓ\|∈\{1,3\}\|W\_\{\\ell\}\|\\in\\\{1,3\\\}\. By our observations in the previous paragraph, it follows thatϵi=ϵj\\epsilon\_\{i\}=\\epsilon\_\{j\}\. In other words, no cancellation among terms of degreekik\_\{i\}can occur in the summation above\. It follows that the Moody polynomial ofΦ\\Phiis not equal to zero, and therefore the Burau representation ofB3B\_\{3\}is faithful\. ∎

#### The parity condition\.

We note that the only information that we used in the proof of Theorem[4\.1](https://arxiv.org/html/2607.05283#S4.Thmtheorem1)was the fact that disks containing an odd number of punctures correspond to a change of sign of corresponding coefficients in the Moody polynomial, while disks containing an even number of punctures preserve the sign of the corresponding coefficients\. We will record this key observation as a lemma\.

To make this precise, letqiq\_\{i\}be a point of intersection of the oriented arcsα\\alphaandβ\\beta\. As in Section[2](https://arxiv.org/html/2607.05283#S2), we letϵi\\epsilon\_\{i\}denote the algebraic intersection numberi^​\(β,α\)\\hat\{i\}\(\\beta,\\alpha\)atqiq\_\{i\}\. IfΔ\\Deltais a disk in the sequence forΦ\\Phicorresponding to the pair of pointsqi,qi\+1∈α∩β∈∂Δq\_\{i\},q\_\{i\+1\}\\in\\alpha\\cap\\beta\\in\\partial\\Delta, then we will say thatΔ\\Deltaissign\-changingifϵi=−ϵi\+1\\epsilon\_\{i\}=\-\\epsilon\_\{i\+1\}andsign\-preservingotherwise\. Let𝒫​\(Δ\)\\mathcal\{P\}\(\\Delta\)denote the set of marked points inDnD\_\{n\}that are contained in the interior ofΔ\\Delta\. We will say that a braidΦ\\Phisatisfies theparity conditionif a diskΔ\\Deltain its disk sequence is sign\-changing if and only if\|𝒫​\(Δ\)\|\|\\mathcal\{P\}\(\\Delta\)\|is odd\.

In other words, our proof of Theorem[4\.1](https://arxiv.org/html/2607.05283#S4.Thmtheorem1)reduces to showing that every 3\-strand braid satisfies the parity condition\. Our next lemma shows that the parity condition gives a sufficient criterion for a braidΦ∈Bn\\Phi\\in\\operatorname\{B\}\_\{n\}to not be contained in the kernel of the Burau representationρn\\rho\_\{n\}\.

Recall from Section[2](https://arxiv.org/html/2607.05283#S2)that the Moody polynomial of a braidϕ\\phiis given by

𝕄ϕ=∑i=1mϵi​tki\\displaystyle\\mathbb\{M\}\_\{\\phi\}=\\sum\_\{i=1\}^\{m\}\\epsilon\_\{i\}t^\{k\_\{i\}\}where eachϵi​tki\\epsilon\_\{i\}t^\{k\_\{i\}\}is the monomial associated to theii\-th crossing ofα\\alphaandβ=\(β∗3\)​ϕ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\phi, and wherem=ι​\(α,β\)m=\\iota\(\\alpha,\\beta\)\. We say that the Moody polynomial ofϕ\\phiadmits no cancellationsif there is no pair of indicesi,ji,jsuch thatki=kjk\_\{i\}=k\_\{j\}andϵi=−ϵj\\epsilon\_\{i\}=\-\\epsilon\_\{j\}in the above expression\.

###### Lemma 4\.2\.

Letϕ∈Bn\\phi\\in B\_\{n\}\. Ifϕ\\phisatisfies the parity condition, then the Moody polynomial ofϕ\\phiadmits no cancellations\.

###### Proof\.

By Corollary[3\.2](https://arxiv.org/html/2607.05283#S3.Thmtheorem2), ifki=kjk\_\{i\}=k\_\{j\}then\|𝒫​\(Δi\)\|\+⋯\+\|𝒫​\(Δj−1\)\|=0\|\\mathcal\{P\}\(\\Delta\_\{i\}\)\|\+\\dots\+\|\\mathcal\{P\}\(\\Delta\_\{j\-1\}\)\|=0\. In particular this sum is even\. However, by the assumption that a given disk in the disk sequence is sign\-changing if and only if it contains an odd number of punctures, this means there are an even number of sign\-changing disks in the subsequenceΔi,…,Δj−1\\Delta\_\{i\},\\dots,\\Delta\_\{j\-1\}\. In particular we haveϵi=ϵj\\epsilon\_\{i\}=\\epsilon\_\{j\}and so this completes the proof\. ∎

In general,nn\-strand braids do not satisfy the parity condition whenn≥4n\\geq 4\. Our strategy for proving the Main Theorem will be to identify certain 4\-braids satisfying the parity condition\.

## 5Push\-maps and minimal position

As discussed in Section[1](https://arxiv.org/html/2607.05283#S1), in order to prove our Main Theorem, it suffices to establish thatρ4\\rho\_\{4\}is faithful when restricted to the subgroup ofB4B\_\{4\}consisting of point\-pushing maps, also known simply as push\-maps\. In light of Lemma[4\.2](https://arxiv.org/html/2607.05283#S4.Thmtheorem2), our strategy will be to identify elements of this subgroup satisfying the parity condition\. In this section, as a first step in that direction, we will characterize configurations of the various arcs and curves involved that lead to the creation of bigons betweenα\\alphaand the image ofβ∗3\\beta\_\{\*\}^\{3\}under the action of push\-maps\.

For the purpose of defining push\-maps, it is convenient to use the notationMod⁡\(Dn\)\\operatorname\{Mod\}\(D\_\{n\}\)to denote the braid groupBn\\operatorname\{B\}\_\{n\}\. Letp∈Dnp\\in D\_\{n\}be one of the marked points inDnD\_\{n\}, and letMod⁡\(Dn,p\)\\operatorname\{Mod\}\(D\_\{n\},p\)denote the subgroup ofMod⁡\(Dn\)\\operatorname\{Mod\}\(D\_\{n\}\)that fixespp\. There is a forgetful mapMod⁡\(Dn,p\)→Mod⁡\(Dn−1\)\\operatorname\{Mod\}\(D\_\{n\},p\)\\to\\operatorname\{Mod\}\(D\_\{n\-1\}\), and the Birman exact sequence forDnD\_\{n\}identifies the kernel of this map withπ1​\(Dn−1,p\)\\pi\_\{1\}\(D\_\{n\-1\},p\):

1→π1​\(Dn−1,p\)→P​u​s​hMod⁡\(Dn,p\)→F​o​r​g​e​tMod⁡\(Dn−1\)→1\\displaystyle 1\\xrightarrow\{\}\\pi\_\{1\}\(D\_\{n\-1\},p\)\\xrightarrow\{Push\}\\operatorname\{Mod\}\(D\_\{n\},p\)\\xrightarrow\{Forget\}\\operatorname\{Mod\}\(D\_\{n\-1\}\)\\xrightarrow\{\}1Given a loopΓ\\Gammainπ1​\(Dn−1,p\)\\pi\_\{1\}\(D\_\{n\-1\},p\), we can consider thepush\-mapobtained by pushing the pointppalongΓ\\Gamma, and the image ofπ1​\(Dn−1,p\)\\pi\_\{1\}\(D\_\{n\-1\},p\)inMod⁡\(Dn,p\)⊂Mod⁡\(Dn\)\\operatorname\{Mod\}\(D\_\{n\},p\)\\subset\\operatorname\{Mod\}\(D\_\{n\}\)is an example of apoint\-pushing subgroup\. We refer the reader to Farb\-Margalit’s book for a detailed discussion of point\-pushing maps and subgroups\[[FM12](https://arxiv.org/html/2607.05283#bib.bib87), Section 4\.2\.2\]\.

Now, letKi≅π1​\(Dn−1,pi\)K\_\{i\}\\cong\\pi\_\{1\}\(D\_\{n\-1\},p\_\{i\}\)denote the point\-pushing subgroup ofBn\\operatorname\{B\}\_\{n\}corresponding to the marked pointpip\_\{i\}in the diskDnD\_\{n\}; see Figure[5\.1](https://arxiv.org/html/2607.05283#S5.F1)for an example of an element of the point\-pushing groupK4K\_\{4\}inB4\\operatorname\{B\}\_\{4\}and its effect on the arcβ∗3\\beta\_\{\*\}^\{3\}\. We remind the reader that throughout the paper, our figures show the arcα\\alphain blue and arcs of the formβ=Φ​\(β∗3\)\\beta=\\Phi\(\\beta\_\{\*\}^\{3\}\)in red\. Henceforth we will also show any loops representing push\-maps in gold\.

\\begin\{overpic\}\[width=433\.62pt\]\{cp\-example\.png\} \\put\(5\.0,13\.5\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(17\.0,1\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta\_\{\*\}^\{3\}$\}\} \\par\\put\(58\.0,13\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(68\.0,1\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\end\{overpic\}Figure 5\.1:The gold loop represents an element of the subgroupK4K\_\{4\}\. The imageβ\\betaofβ∗3\\beta\_\{\*\}^\{3\}under the push\-map around the gold loop is shown in red on the right\.In this part of the paper, we do not need to restrict to the casen=4n=4, and so we will work in the diskDnD\_\{n\}, the braid groupBn\\operatorname\{B\}\_\{n\}, and its subgroupKnK\_\{n\}forn≥4n\\geq 4\. Before we begin, we note the following\.

- •Without loss of generality, we can always choose a representative of a loopΓ\\Gammathat intersects itself minimally and transversely, and hence there are finitely many points of self\-intersection\.
- •The results in this section hold if we replaceKnK\_\{n\}with any other point\-pushing subgroupKjK\_\{j\}wherej≥4j\\geq 4\.
- •In this section we do not assume that our push\-maps necessarily correspond to simple loops\. However, we only require simple loops in order to prove Theorem[6\.6](https://arxiv.org/html/2607.05283#S6.Thmtheorem6)below\.

#### Bigon\-forming polygons\.

LetΓ∈Kn\\Gamma\\in K\_\{n\}, and letβ\\betabe any simple arc joiningp∗p\_\{\*\}top3p\_\{3\}\. We wish to identify local configurations of subarcs ofα,β\\alpha,\\beta, andΓ\\Gammathat lead to the creation of a bigon between\(β\)​Γ\(\\beta\)\\Gammaandα\\alpha\. As a warm\-up, suppose that as we travel alongΓ\\Gammathere is a subarc ofβ\\betathat precedes a bigon betweenΓ\\Gammaandα\\alpha\. Then pushing alongΓ\\Gammawill create a bigon \(in fact, two\); see Figure[2\(a\)](https://arxiv.org/html/2607.05283#S5.F2.sf1)\.

![Refer to caption](https://arxiv.org/html/2607.05283v1/bigon-push.png)\(a\)A bigon betweenΓ\\Gamma\(gold\) andα\\alpha\(blue\)\.
\\begin\{overpic\}\[width=390\.25534pt\]\{fake\-bigon\.png\} \\put\(40\.0,2\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(6\.0,10\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma^\{\(2\)\}$\}\} \\put\(30\.0,10\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma^\{\(1\)\}$\}\} \\end\{overpic\}\(b\)A generalized bigon of length 2 betweenΓ\\Gammaandα\\alpha\.
Figure 5\.2:Of course, in general we can avoid such a scenario simply by assuming that all curves are in pairwise minimal position\. However, this points us to a more general kind of phenomenon in the case whereΓ\\Gammais not simple: a sequence of subarcs ofΓ\\Gammathat, taken together, effectively form a bigon withΓ\\Gamma, as shown in Figure[2\(b\)](https://arxiv.org/html/2607.05283#S5.F2.sf2)\. Then, depending on the ordering of these subarcs \(as determined by the orientation ofΓ\\Gamma\), pushing alongΓ\\Gammacan similarly create a bigon\.

To make this more precise, given two subarcsΓ\(1\),Γ\(2\)\\Gamma^\{\(1\)\},\\Gamma^\{\(2\)\}ofΓ\\GammainDnD\_\{n\}with disjoint interiors, we will say thatΓ\(1\)\\Gamma^\{\(1\)\}precedesΓ\(2\)\\Gamma^\{\(2\)\}and writeΓ\(1\)<Γ\(2\)\\Gamma^\{\(1\)\}<\\Gamma^\{\(2\)\}if the initial point ofΓ\(1\)\\Gamma^\{\(1\)\}precedes the initial point ofΓ\(2\)\\Gamma^\{\(2\)\}as we trace outΓ\\Gamma\. We will further refer to a union of subarcsΓ\(1\),…,Γ\(ℓ\)\\Gamma^\{\(1\)\},\\ldots,\\Gamma^\{\(\\ell\)\}that are pairwise disjoint as apiecewiseΓ\\Gamma\-arc of lengthℓ\\ellif the following conditions are satisfied:

- •for eachi∈\{1,…,ℓ−1\}i\\in\\\{1,\\ldots,\\ell\-1\\\}, the final point ofΓ\(i\)\\Gamma^\{\(i\)\}is a self\-intersection point ofΓ\\GammainDnD\_\{n\}, such that the final point ofΓ\(i\)\\Gamma^\{\(i\)\}is the initial point ofΓ\(i\+1\)\\Gamma^\{\(i\+1\)\};
- •Γ\(1\)<Γ\(2\)<⋯<Γ\(ℓ\)\\Gamma^\{\(1\)\}<\\Gamma^\{\(2\)\}<\\cdots<\\Gamma^\{\(\\ell\)\}; and
- •the unionΓ\(1\)∪⋯∪Γ\(ℓ\)\\Gamma^\{\(1\)\}\\cup\\cdots\\cup\\Gamma^\{\(\\ell\)\}inDnD\_\{n\}contains no loops\.

We will now identify three types of local configurations \(in addition to bigons\) betweenα\\alpha,β\\beta, andΓ\\Gammathat lead to the creation of bigons betweenα\\alphaand\(β\)​Γ\(\\beta\)\\Gamma\.

1. 1\.Generalized bigons\.Consider now a piecewiseΓ\\Gamma\-arcγ\\gammaof lengthℓ\\ellwhere the initial point ofΓ\(1\)\\Gamma^\{\(1\)\}and the final point ofΓ\(ℓ\)\\Gamma^\{\(\\ell\)\}are both points of intersection ofΓ\\Gammawithα\\alpha, so thatγ\\gammatogether with theα\\alpha\-subarc with these two endpoints forms a\(ℓ\+1\)\(\\ell\+1\)\-gon inDnD\_\{n\}containing no marked points in its interior\. We will refer to such an\(ℓ\+1\)\(\\ell\+1\)\-gon as ageneralized bigon\. Figure[2\(b\)](https://arxiv.org/html/2607.05283#S5.F2.sf2)shows a generalized bigon withℓ=2\\ell=2\. If the piecewiseΓ\\Gamma\-arc in a generalized bigon is preceded by a point of intersection withβ\\beta, then\(β\)​Γ\(\\beta\)\\Gammaandα\\alphawill form a bigon in the same way as if the piecewiseΓ\\Gamma\-arc were replaced by a single subarc ofΓ\\Gamma\. We emphasize that the orientation and ordering of theΓ\\Gamma\-subarcs as we traverseΓ\\Gammais crucial here, since otherwiseΓ\\Gammawould not push theβ\\beta\-subarc all the way around to form a bigon between\(β\)​Γ\(\\beta\)\\Gammaandα\\alpha\. Finally, a “standard” bigon is a generalized bigon with a piecewiseΓ\\Gamma\-arcγ\\gammaof length 1\.
2. 2\.Generalized trigons\.Figure[3\(a\)](https://arxiv.org/html/2607.05283#S5.F3.sf1)shows another configuration of curves that leads to the creation of a bigon, namely a trigon inD4D\_\{4\}whose boundary consists of a subarc from each of the arcsα\\alphaandβ,\\beta,and the loopΓ\\Gamma, where theΓ\\Gamma\-subarc is oriented from theβ\\beta\-arc to theα\\alpha\-arc\. We will refer to such a configuration of curves as aβ\\beta\-to\-α\\alphatrigon\. The orientation of theΓ\\Gamma\-subarc is crucial here: if its orientation were reversed, then\(β\)​Γ\(\\beta\)\\Gammadoes not form any bigons withα\\alphain this region\. As with bigons, we can replace the single subarc ofΓ\\Gammahere with a piecewiseΓ\\Gamma\-arc of any length; we refer to any such configuration as ageneralizedβ\\beta\-to\-α\\alphatrigon\. See Figure[3\(b\)](https://arxiv.org/html/2607.05283#S5.F3.sf2)for an example\. \\begin\{overpic\}\[width=303\.53267pt\]\{trigon\.png\} \\put\(65\.0,30\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\put\(25\.0,30\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma$\}\} \\put\(5\.0,16\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\end\{overpic\}\(a\)Aβ\\beta\-to\-α\\alphatrigon\.\\begin\{overpic\}\[width=377\.24727pt\]\{fake\-trigon\.png\} \\put\(67\.0,14\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\put\(55\.0,42\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma^\{\(1\)\}$\}\} \\put\(30\.0,41\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma^\{\(2\)\}$\}\} \\put\(18\.0,14\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma^\{\(3\)\}$\}\} \\put\(4\.0,10\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\end\{overpic\}\(b\)A five\-sided example of a generalizedβ\\beta\-to\-α\\alphatrigon\. Figure 5\.3:
3. 3\.Generalized rectangles\.Another configuration of curves that may create bigons betweenα\\alphaand\(β\)​Γ\(\\beta\)\\Gammaarises when two parallel subarcs ofΓ\\Gammatraveling betweenβ\\betaandα\\alphaform a rectangle containing no marked points; we will call this aβ\\beta\-to\-α\\alpharectangle\. As with bigons and trigons, we can use two piecewiseΓ\\Gamma\-arcs instead of twoΓ\\Gamma\-arcs, and we refer to any such configuration as ageneralized rectangle; see Figure[4\(a\)](https://arxiv.org/html/2607.05283#S5.F4.sf1)for an example\. We emphasize that in the case of a generalized rectangle, we do not necessarily require the two piecewiseΓ\\Gamma\-arcs to be disjoint\. In other words, we allow “degenerate” generalized rectangles, in which some number of theΓ\\Gamma\-segments involved can play a role in both piecewiseΓ\\Gamma\-arcs; we will see examples of this in the proof of the next lemma \(see Figure[4\(b\)](https://arxiv.org/html/2607.05283#S5.F4.sf2)\)\. Note also that every generalized rectangle includes two crossings ofΓ\\Gammaandα\\alphawith the same sign that are adjacent, as the crossings are ordered alongα\\alpha\. \\begin\{overpic\}\[width=303\.53267pt\]\{non\-alt\-trigon\.png\} \\put\(38\.0,13\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma$\}\} \\put\(2\.0,28\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(98\.0,28\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\end\{overpic\}\(a\)An example of a generalized rectangle with piecewiseΓ\\Gamma\-arcs of length 1 and length 2\.\\begin\{overpic\}\[width=303\.53267pt\]\{case2\-r2\.png\} \\put\(11\.0,12\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\put\(13\.0,45\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(25\.0,15\.0\)\{$\\Gamma^\{\(1\)\}$\} \\put\(32\.0,26\.0\)\{$\\Gamma^\{\(2\)\}$\} \\put\(26\.0,40\.0\)\{$\\Gamma^\{\(3\)\}$\} \\put\(82\.0,64\.0\)\{$\\Gamma^\{\(4\)\}$\} \\end\{overpic\}\(b\)A degenerate generalized rectangle formed by ​Γ\(1\)∪Γ\(2\)∪Γ\(3\)\\mbox\{\\qquad\}\\Gamma^\{\(1\)\}\\cup\\Gamma^\{\(2\)\}\\cup\\Gamma^\{\(3\)\}andΓ\(1\)∪Γ\(2\)∪Γ\(4\)\\Gamma^\{\(1\)\}\\cup\\Gamma^\{\(2\)\}\\cup\\Gamma^\{\(4\)\}\. Figure 5\.4:

In the proof of the next lemma, we will see that generalized bigons,β\\beta\-to\-α\\alphatrigons, and rectangles are the only types of configurations that give rise to bigons between\(β\)​Γ\(\\beta\)\\Gammaandα\\alpha\. Hence we will refer to these configurations collectively asbigon\-forming polygons\.

#### Proper products\.

LetΦ∈Kn\\Phi\\in K\_\{n\}, and letβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phi\. Our next step will be to establish a set of conditions on a loopΓ\\Gammabased atpnp\_\{n\}that will imply\(β\)​Γ\(\\beta\)\\Gammaandα\\alphaare in minimal position\. To that end, we letΓI\\Gamma\_\{I\}denote theinitial componentofΓ\\Gamma, that is, the subarc ofΓ\\Gammafrompnp\_\{n\}to the first crossing ofΓ\\Gammawithα\\alpha\. Similarly, we letΓF\\Gamma\_\{F\}denote thefinal componentofΓ\\Gamma, traveling from its final crossing withα\\alphaback topnp\_\{n\}\. We also letβ′\\beta^\{\\prime\}denote theβ\\beta\-component of the innermost disk in the disk sequence ofΦ\\Phithat contains onlypnp\_\{n\}if such a disk exists; otherwise we setβ′=∅\\beta^\{\\prime\}=\\emptyset\. We say thatΦ⋅Γ\\Phi\\cdot\\Gammais aproper productif we can choose a representative of the loopΓ\\Gammathat is in minimal position with respect toα\\alphaandβ\\beta, and satisfying the following two conditions:

1. 1\.there are no bigon\-forming polygons betweenα\\alpha,β\\beta, andΓ\\Gamma; and
2. 2\.one of the following two conditions holds: 1. \(a\)ΓI∩β′≠∅\\Gamma\_\{I\}\\cap\\beta^\{\\prime\}\\neq\\emptyset; or 2. \(b\)ΓI∩β′=∅\\Gamma\_\{I\}\\cap\\beta^\{\\prime\}=\\emptysetandΓF∩β′=∅\\Gamma\_\{F\}\\cap\\beta^\{\\prime\}=\\emptyset\.

The second condition ensures that postcomposing withΓ\\Gammadoes not “undo” the creation \(byΦ\\Phi\) of a disk containing onlyp4p\_\{4\}\. Figure[5\.5](https://arxiv.org/html/2607.05283#S5.F5)depicts the local picture for each type of configuration: in first case \(on the left in Figure[5\.5](https://arxiv.org/html/2607.05283#S5.F5)\), reversing the orientation ofΓ\\Gammahas the effect of swapping the roles ofΓI\\Gamma\_\{I\}andΓF\\Gamma\_\{F\}, resulting in a loop that does not form a proper product\. In practice, we will generally be given a push\-mapΦ\\Phi, and then seek the second factorΓ\\Gammato form a proper product withΦ\\Phi\. We will also refer toΦ⋅Γ1⋅Γ2\\Phi\\cdot\\Gamma\_\{1\}\\cdot\\Gamma\_\{2\}as a proper product if bothΦ⋅Γ1\\Phi\\cdot\\Gamma\_\{1\}and\(Φ⋅Γ1\)⋅Γ2\(\\Phi\\cdot\\Gamma\_\{1\}\)\\cdot\\Gamma\_\{2\}are proper products\.

\\begin\{overpic\}\[width=390\.25534pt\]\{proper\-fact\.png\} \\put\(40\.0,18\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta^\{\\prime\}$\}\} \\put\(93\.0,18\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta^\{\\prime\}$\}\} \\put\(44\.5,5\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma\_\{I\}$\}\} \\put\(5\.0,11\.5\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(59\.0,11\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\end\{overpic\}Figure 5\.5:Two examples of the local picture for Condition 2 of a proper productΦ⋅Γ\\Phi\\cdot\\Gammaindicating the initial and final components ofΓ\\Gammain each case; reversing the orientation ofΓ\\Gammain the left\-hand figure gives a non\-example\.The next lemma gives a straightforward criterion for ensuring that the product of two push\-maps does not yield any bigons betweenα\\alphaand the image ofβ∗3\\beta\_\{\*\}^\{3\}under the product\.

###### Proposition 5\.1\.

LetΦ∈Kn\\Phi\\in K\_\{n\}, and letβ=Φ​\(β∗3\)\\beta=\\Phi\(\\beta\_\{\*\}^\{3\}\)\. IfΓ\\Gammais a loop based atpnp\_\{n\}such thatΦ⋅Γ\\Phi\\cdot\\Gammais a proper product, then the arcs\(β\)​Γ\(\\beta\)\\Gammaandα\\alphaare in minimal position\.

###### Proof\.

We begin by choosing a specific representative ofΓ\\Gammaso thatα,β\\alpha,\\betaandΓ\\Gammaare pairwise in minimal position with no triple points of intersection\. Up to isotopy we can assume that\(β\)​Γ\(\\beta\)\\Gammaandα\\alphaare transverse inDnD\_\{n\}\. Consider now a points∈\(β\)​Γ∩αs\\in\(\\beta\)\\Gamma\\cap\\alpha\. Ifssalso lies inβ∩α\\beta\\cap\\alphawe will refer to this as anoriginalpoint of intersection; otherwise we say thatssis anewpoint of intersection\.

Suppose now thatα\\alphaand\(β\)​Γ\(\\beta\)\\Gammaform a bigon, and suppose further that the two vertices of this bigon are both new points of intersection\. New points of intersection occur when aβ\\beta\-subarc precedes anα\\alpha\-subarc as we travel alongΓ\\Gamma\. Therefore our two vertices correspond to adjacent crossings ofΓ\\Gammaandα\\alphaof the same sign or of different signs, as shown in Figure[5\.6](https://arxiv.org/html/2607.05283#S5.F6)\.

\\begin\{overpic\}\[width=260\.17464pt\]\{bigon\-signs\.png\} \\put\(48\.0,1\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(47\.0,15\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\(\\beta\)\\Gamma$\}\} \\put\(58\.0,\-1\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma$\}\} \\end\{overpic\}Figure 5\.6:On the left two subarcs ofΓ\\Gammaintersectβ\\betawith the same sign; on the right two subarcs ofΓ\\Gammaintersectβ\\betawith different signs\.If they are the same sign, this implies the existence of a generalized rectangle betweenΓ\\Gamma,α\\alpha, andβ\\beta, which contradicts our assumption thatΦ⋅Γ\\Phi\\cdot\\Gammais a proper product\. If they are different signs, this implies the existence of a generalized bigon betweenΓ\\Gammaandα\\alpha, which again contradicts our assumption thatΦ⋅Γ\\Phi\\cdot\\Gammais a proper product\. Therefore at least one vertex of any bigon betweenα\\alphaand\(β\)​Γ\(\\beta\)\\Gammamust be an original point of intersection\.

Now, the support of the push\-mapΓ\\Gammais a regular neighborhood ofΓ\\GammainDnD\_\{n\}\. Hence we can decompose\(β\)​Γ\(\\beta\)\\Gammaas the union of subarcs ofβ\\betathat are fixed byΓ\\Gamma, which we refer to asoriginal subarcs, together withnew subarcsthat lie in the support ofΓ\\Gamma\. Each of the new subarcs is one of four types: \(I\) joins an original subarc to a new point of intersection, \(II\) joins two new points of intersection forming part of the boundary of a disk boundingpnp\_\{n\}, \(III\) joins two original subarcs, or \(IV\) joins two new points of intersection in a disk not containing onlypnp\_\{n\}\.

\\begin\{overpic\}\[width=390\.25534pt\]\{type1\-4\.png\} \\put\(13\.0,36\.0\)\{\(II\)\} \\put\(20\.5,42\.5\)\{\(I\)\} \\put\(76\.0,42\.5\)\{\(III\)\} \\put\(4\.0,10\.0\)\{\(IV\)\} \\end\{overpic\}Figure 5\.7:Subarcs of types \(I\)\-\(IV\): here original subarcs ofβ\\betaappear in red, while new subarcs ofβ\\betaare colored purple\.Suppose now that the verticess1,s2s\_\{1\},s\_\{2\}of our bigon are both original points of intersection\. Then the subarc of\(β\)​Γ\(\\beta\)\\Gammathat joinss1s\_\{1\}tos2s\_\{2\}cannot contain any new subarcs of type \(I\), type \(II\), or type \(III\) and must therefore contain a new subarc of type \(IV\)\. It follows that we have two subarcsβ′⊂β\\beta^\{\\prime\}\\subset\\betaandα′⊂α\\alpha^\{\\prime\}\\subset\\alphasatisfying the following three statements:

- •the unionα′∪β′\\alpha^\{\\prime\}\\cup\\beta^\{\\prime\}does not form a bigon;
- •the unionα′∪\(β′\)​Γ\\alpha^\{\\prime\}\\cup\(\\beta^\{\\prime\}\)\\Gammaforms a bigon; and
- •the intersectionα′∩\(β′\)​Γ=α′∩β′=\{s1,s2\}\\alpha^\{\\prime\}\\cap\(\\beta^\{\\prime\}\)\\Gamma=\\alpha^\{\\prime\}\\cap\\beta^\{\\prime\}=\\\{s\_\{1\},s\_\{2\}\\\}\.

The disk bounded byα′∪β′\\alpha^\{\\prime\}\\cup\\beta^\{\\prime\}inDnD\_\{n\}must contain at least one marked point, since we are assuming thatα\\alphaandβ\\betaare in minimal position\. If the disk bounded byα′∪β′\\alpha^\{\\prime\}\\cup\\beta^\{\\prime\}containedpip\_\{i\}for somei≠ni\\neq n, then the disk bounded byα′∪\(β′\)​Γ\\alpha^\{\\prime\}\\cup\(\\beta^\{\\prime\}\)\\Gammawould also containpip\_\{i\}, but this is not possible sinceα′∪\(β′\)​Γ\\alpha^\{\\prime\}\\cup\(\\beta^\{\\prime\}\)\\Gammabounds a bigon\. Therefore the disk bounded byα′∪β′\\alpha^\{\\prime\}\\cup\\beta^\{\\prime\}contains only the marked pointpnp\_\{n\}\. We must also have that the final componentΓF\\Gamma\_\{F\}ofΓ\\Gammaintersectsβ′\\beta^\{\\prime\}in a single point, sinceα′∪\(β′\)​Γ\\alpha^\{\\prime\}\\cup\(\\beta^\{\\prime\}\)\\Gammadoes not containpnp\_\{n\}\. Moreover, no other subarc ofΓ\\Gammacan intersectβ′\\beta^\{\\prime\}, as otherwiseα′∪\(β′\)​Γ\\alpha^\{\\prime\}\\cup\(\\beta^\{\\prime\}\)\\Gammawould not form a bigon\. In particular, it follows thatΓI\\Gamma\_\{I\}intersectsα′∪\(β′\)​Γ\\alpha^\{\\prime\}\\cup\(\\beta^\{\\prime\}\)\\Gammain a single point inα′\\alpha^\{\\prime\}\. Hence the loopΓ\\Gammamust intersect the disk bounded byα′∪β′\\alpha^\{\\prime\}\\cup\\beta^\{\\prime\}as shown in Figure[5\.8](https://arxiv.org/html/2607.05283#S5.F8)\. However, such a configuration contradicts the fact thatΦ⋅Γ\\Phi\\cdot\\Gammais a proper product\. Therefore the verticess1s\_\{1\}ands2s\_\{2\}cannot both be original points of intersection\.

\\begin\{overpic\}\[width=346\.89731pt\]\{trigon\-tail\.png\} \\put\(3\.0,10\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha^\{\\prime\}$\}\} \\put\(2\.5,1\.5\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta^\{\\prime\}$\}\} \\put\(\-1\.0,6\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$s\_\{1\}$\}\} \\put\(24\.0,6\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$s\_\{2\}$\}\} \\put\(45\.5,6\.5\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma$\}\} \\put\(18\.0,0\.0\)\{\{\\color\[rgb\]\{0,0\.66796875,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.66796875,0\}$\\Gamma\_\{F\}$\}\} \\put\(16\.0,6\.5\)\{\{\\color\[rgb\]\{1,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,1\}$\\Gamma\_\{I\}$\}\} \\put\(10\.0,5\.5\)\{\{$p\_\{n\}$\}\} \\end\{overpic\}Figure 5\.8:A new bigon involving two original points of intersections1s\_\{1\}ands2s\_\{2\}betweenα\\alphaandβ\\betamust arise from a configuration of curves as depicted\.It remains to consider the case in which precisely one vertex of our bigon betweenα\\alphaand\(β\)​Γ\(\\beta\)\\Gammais an original point of intersection; we denote this vertex bys1s\_\{1\}\. The other vertexs2s\_\{2\}must therefore be a new point of intersection formed whenΓ\\Gammameets a subarc ofβ\\betaand pushes it along until it crossesα\\alpha\. Without loss of generality, we can assume thats2s\_\{2\}is the new point of intersection to the left of the loopΓ\\Gamma, as shown in Figure[5\.9](https://arxiv.org/html/2607.05283#S5.F9)\.

\\begin\{overpic\}\[width=260\.17464pt\]\{one\-point\-case\-1\.png\} \\put\(12\.0,1\.5\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma$\}\} \\put\(6\.0,8\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\put\(56\.0,0\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(56\.0,11\.0\)\{\\small\{$s\_\{2\}$\}\} \\put\(94\.0,11\.0\)\{\\small\{$s\_\{2\}$\}\} \\put\(87\.0,18\.5\)\{\\small\{$s\_\{1\}$\}\} \\end\{overpic\}Figure 5\.9:The simplest case where a bigon betweenα\\alphaand\(β\)​Γ\(\\beta\)\\Gammainvolves precisely one original point of intersection\.Then there are two possible candidate regions for a bigon withs2s\_\{2\}as a vertex: one to the left ofα\\alphaas shown in the figure, and one to the right\. In each case, the vertexs1s\_\{1\}lies alongα\\alphaon the same side ofΓ\\Gammaass2s\_\{2\}in the local picture shown in Figure[5\.9](https://arxiv.org/html/2607.05283#S5.F9)\. In the first case, there must be a piecewiseΓ\\Gamma\-arc forming a generalizedβ\\beta\-to\-α\\alphatrigon withs2s\_\{2\}corresponding to its final point of intersection withα\\alpha, which contradicts thatΦ⋅Γ\\Phi\\cdot\\Gammais a proper product; see Figure[5\.10](https://arxiv.org/html/2607.05283#S5.F10)\.

\\begin\{overpic\}\[width=303\.53267pt\]\{case2\-r1\.png\} \\put\(23\.0,35\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(13\.0,17\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\put\(97\.0,9\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma$\}\} \\end\{overpic\}Figure 5\.10:A possible bigon that arises from a generalized trigon\. The oriented arcs in gold are subarcs of a loopΓ\\Gamma\.The second case also cannot occur, since this would imply that\(β\)​Γ\(\\beta\)\\Gammacontinues on along a new subarc froms2s\_\{2\}tos1s\_\{1\}to form our bigon, which in turn implies thatΓ\\Gammaitself must form a generalized bigon withβ\\betaand thats1s\_\{1\}is also a new point of intersection, both of which are contradictions\. The proposition follows\. ∎

In the next section, we will restrict our attention to the casen=4n=4\.

## 6Disk sequences in the 4\-strand braid group

The aim of this section is to prove Theorem[6\.6](https://arxiv.org/html/2607.05283#S6.Thmtheorem6)below, which states that the Burau representationρ4\\rho\_\{4\}is faithful on its restriction to the point\-pushing groupK4K\_\{4\}\. By Proposition[1\.2](https://arxiv.org/html/2607.05283#S1.Thmtheorem2), this will imply our Main Theorem\. Our strategy for proving Theorem[6\.6](https://arxiv.org/html/2607.05283#S6.Thmtheorem6)is to use proper products to identify certain braids that satisfy the parity condition\. This will allow us to apply Moody’s theorem \(Theorem[2\.3](https://arxiv.org/html/2607.05283#S2.Thmtheorem3)\) to establish faithfulness\.

### 6\.1Disks and parity in𝐁4\{\\bm\{\\operatorname\{B\}\}\_\{4\}\}

We begin with a careful analysis of the types of disks that can arise in the disk sequence of a 4\-strand braid\. To that end, we introduce some useful notation: for a diskΔ\\Deltacontained inD4\\operatorname\{D\}\_\{4\}, we let𝒫​\(Δ\)\\mathcal\{P\}\(\\Delta\)denote the set of marked points that are contained in the interior ofΔ\\Delta, and hence𝒫​\(Δ\)⊆\{p1,p2,p3,p4\}\\mathcal\{P\}\(\\Delta\)\\subseteq\\\{p\_\{1\},p\_\{2\},p\_\{3\},p\_\{4\}\\\}\. We note that up to this point in the paper we have not used the assumption thatn=4n=4\. We will use that assumption in this section, particularly in Lemmas[6\.1](https://arxiv.org/html/2607.05283#S6.Thmtheorem1)and[6\.3](https://arxiv.org/html/2607.05283#S6.Thmtheorem3)\.

###### Lemma 6\.1\.

LetΔ\\Deltabe a disk arising in the disk sequence of an element ofB4\\operatorname\{B\}\_\{4\}\. ThenΔ\\Deltais sign\-preserving if and only if𝒫​\(Δ\)\\mathcal\{P\}\(\\Delta\)is one of the following:

\{p1,p3\},\{p1,p4\},\{p2,p3\},\{p2,p4\},\{p1,p3,p4\},\{p2,p3,p4\}\.\\\{p\_\{1\},p\_\{3\}\\\},\\hskip 7\.22743pt\\\{p\_\{1\},p\_\{4\}\\\},\\hskip 7\.22743pt\\\{p\_\{2\},p\_\{3\}\\\},\\hskip 7\.22743pt\\\{p\_\{2\},p\_\{4\}\\\},\\hskip 7\.22743pt\\\{p\_\{1\},p\_\{3\},p\_\{4\}\\\},\\hskip 7\.22743pt\\\{p\_\{2\},p\_\{3\},p\_\{4\}\\\}\.Moreover,Δ\\Deltais sign\-changing if and only if𝒫​\(Δ\)\\mathcal\{P\}\(\\Delta\)is one of the following:

\{p3\},\{p4\},\{p3,p4\},\{p1,p2,p3\},\{p1,p2,p4\},\{p1,p2,p3,p4\}\.\\\{p\_\{3\}\\\},\\hskip 7\.22743pt\\\{p\_\{4\}\\\},\\hskip 7\.22743pt\\\{p\_\{3\},p\_\{4\}\\\},\\hskip 7\.22743pt\\\{p\_\{1\},p\_\{2\},p\_\{3\}\\\},\\hskip 7\.22743pt\\\{p\_\{1\},p\_\{2\},p\_\{4\}\\\},\\hskip 7\.22743pt\\\{p\_\{1\},p\_\{2\},p\_\{3\},p\_\{4\}\\\}\.

###### Proof\.

LetΔ\\Deltabe a disk in the disk sequence of an elementΦ∈B4\\Phi\\in\\operatorname\{B\}\_\{4\}, and letqqandq′q^\{\\prime\}denote the two points of intersection betweenα\\alphaandβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phithat lie in∂Δ\\partial\\Delta\. Without loss of generality, we assume thatqqlies betweenp1p\_\{1\}andq′q^\{\\prime\}onα\\alpha, and thatqqprecedesq′q^\{\\prime\}as we traveling alongβ\\beta\.

Suppose that𝒫​\(Δ\)=\{p1\}\\mathcal\{P\}\(\\Delta\)=\\\{p\_\{1\}\\\}\. Sinceβ\\betaandα\\alphaintersect transversely atqq, the subarc ofβ\\betajoining the basepointp∗p\_\{\*\}toqqmust intersect∂Δ\\partial\\Delta\. Hence this subarc either intersects theβ\\betacomponent of∂Δ\\partial\\Deltaor the subarc ofα\\alphawith endpointsqqandq′q^\{\\prime\}\. The former contradicts our assumption thatβ\\betais simple, and the latter contradicts our assumption thatβ\\betaandα\\alphaare in minimal position\. Therefore𝒫​\(Δ\)≠\{p1\}\\mathcal\{P\}\(\\Delta\)\\neq\\\{p\_\{1\}\\\}\. By a similar argument, we see that𝒫​\(Δ\)≠\{p2\}\\mathcal\{P\}\(\\Delta\)\\neq\\\{p\_\{2\}\\\}and that𝒫​\(Δ\)≠\{p1,p2\}\\mathcal\{P\}\(\\Delta\)\\neq\\\{p\_\{1\},p\_\{2\}\\\}\.

We next claim thatΔ\\Deltais sign\-preserving if and only if it contains precisely one of the two endpoints ofα\\alpha; the lemma follows immediately from the claim\. Suppose first thatΔ\\Deltais sign\-preserving\. Without loss of generality, we assume thatβ\\betais oriented down \(in our standard picture whereα\\alphais horizontal\) at bothqqandq′q^\{\\prime\}, and thatΔ\\Deltacontainsp1p\_\{1\}; see Figure[6\.1](https://arxiv.org/html/2607.05283#S6.F1)\.

\\begin\{overpic\}\[width=130\.08731pt\]\{sign\-pres\.png\} \\put\(56\.0,35\.0\)\{$q$\} \\put\(81\.0,12\.0\)\{$q^\{\\prime\}$\} \\end\{overpic\}Figure 6\.1:A sign\-preserving diskThen∂Δ\\partial\\Deltais the union of a subarc ofβ\\betaand a subarc ofα\\alpha, both with endpointsq,q′q,q^\{\\prime\}, and the interior ofΔ\\Deltalies to the right ofβ\\betaas we travel alongβ\\betafromqqtoq′q^\{\\prime\}\. This implies thatΔ\\Deltadoes not contain the subarc ofα\\alphawith endpointsq′q^\{\\prime\}andp2p\_\{2\}; in particularΔ\\Deltadoes not containp2p\_\{2\}\. A similar argument shows that ifΔ\\Deltais sign\-changing and containsp1p\_\{1\}, then it also containsp2p\_\{2\}\. The claim follows, and we are done\. ∎

### 6\.2The point\-pushing subgroup𝑲4\{\\bm\{K\}\_\{4\}\}

Our focus in the remainder of this section will be the point\-pushing subgroupK4K\_\{4\}\. Our next step will be to analyze the types of disks that can appear in the disk sequence of certain braids inK4K\_\{4\}\. Specifically, we will be constructing proper products of certain elements inK4K\_\{4\}to which we can apply Lemma[4\.2](https://arxiv.org/html/2607.05283#S4.Thmtheorem2)\. We will then analyze the corresponding disk sequences, which will ultimately provide a mechanism for applying Theorem[2\.3](https://arxiv.org/html/2607.05283#S2.Thmtheorem3)\.

###### Lemma 6\.2\.

LetΦ∈K4\\Phi\\in K\_\{4\}, letΓ∈K4\\Gamma\\in K\_\{4\}be a push\-map along a simple loop, and suppose thatΦ=Ψ⋅Γ\\Phi=\\Psi\\cdot\\Gammais a proper product\. LetΔ1,…,Δm−1\\Delta\_\{1\},\\dots,\\Delta\_\{m\-1\}denote the disk sequence ofΦ\\Phi\. Then, for any fixedii, eitherΔi\\Delta\_\{i\}is a 1\-disk containing onlyp4p\_\{4\}, or there is an isotopy ofβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phiso that theβ\\beta\-component of the boundaryΔi\\Delta\_\{i\}is disjoint fromΓ\\Gamma\.

Before giving the proof, we emphasize that Lemma[6\.2](https://arxiv.org/html/2607.05283#S6.Thmtheorem2)only guarantees that theβ\\beta\-component of any single diskΔi\\Delta\_\{i\}can be isotoped offΓ\\Gamma\. It does not guarantee that each suchβ\\beta\-component can be isotoped offΓ\\Gammasimultaneously; indeed, this is not possible in general\. We also introduce some notation that will be convenient in the proof of the lemma and in what follows: ifγ\\gammais an arc or a loop in the diskDnD\_\{n\}, we will refer to a disk whose boundary is the union of a single subarc ofα\\alphatogether with a single subarc ofγ\\gammaas anα\\alpha\-γ\\gammadisk\.

###### Proof\.

Letβ′=\(β∗3\)​Ψ\\beta^\{\\prime\}=\(\\beta\_\{\*\}^\{3\}\)\\Psi\. Again, we choose fixed representatives ofΓ\\Gammaandα\\alphathat are in minimal position\. By Proposition[5\.1](https://arxiv.org/html/2607.05283#S5.Thmtheorem1), we may choose a representative ofβ′\\beta^\{\\prime\}so that\(β′\)​Γ\(\\beta^\{\\prime\}\)\\Gammais in minimal position with respect toα\\alpha\. SinceΓ\\Gammais a simple loop , we can represent the action of the push\-mapΓ\\Gammavia the schematic in Figure[6\.2](https://arxiv.org/html/2607.05283#S6.F2)\.

\\begin\{overpic\}\[width=390\.25534pt\]\{local\-picture\.png\} \\put\(32\.0,50\.0\)\{\{\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}$\\alpha$\}\} \\put\(68\.0,50\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\beta$\}\} \\put\(85\.0,37\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\Gamma\_\{1\}$\}\} \\end\{overpic\}\(a\)The local picture of pushing alongΓ1\\Gamma\_\{1\}\.![Refer to caption](https://arxiv.org/html/2607.05283v1/local-picture-pushed.png)\(b\)Pushing the firstβ\\betastrand

Figure 6\.2:Here blue arcs represent subarcs ofα\\alpha, while red/pink arcs represent subarcs ofβ\\beta\.Figure[2\(b\)](https://arxiv.org/html/2607.05283#S6.F2.sf2)shows the resulting local picture forβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phiafter pushingβ′\\beta^\{\\prime\}alongΓ\\Gamma, possibly creating a number of new disks in the associated disk sequence\. For each such disk that does not containp4p\_\{4\}, Figure[2\(b\)](https://arxiv.org/html/2607.05283#S6.F2.sf2)shows that itsβ\\beta\-component is disjoint fromΓ\\Gamma\. For the remainingα\\alpha\-β\\betadisks in Figure[2\(b\)](https://arxiv.org/html/2607.05283#S6.F2.sf2), there is an isotopy ofβ\\betainDnD\_\{n\}that preserves the disk sequence while removing the intersection of theβ\\beta\-component of these disks fromΓ\\Gamma\. This isotopy can be visualized by sliding the “topmost” part of anyβ\\beta\-component of the boundary of a disk that intersectsΓ\\Gammain Figure[2\(b\)](https://arxiv.org/html/2607.05283#S6.F2.sf2)upwards along the rightmost subarc ofα\\alphathat appears in the local picture; one can perform a single isotopy that moves all suchβ\\beta\-subarcs simultaneously offΓ\\Gamma\. ∎

#### A particular push\-map\.

LetΓ1\\Gamma\_\{1\}be the push\-map corresponding to the loop shown in Figure[6\.3](https://arxiv.org/html/2607.05283#S6.F3)\.

![Refer to caption](https://arxiv.org/html/2607.05283v1/gamma1.png)Figure 6\.3:A particular push mapΓ1\\Gamma\_\{1\}\.###### Lemma 6\.3\.

LetΦ∈K4\\Phi\\in K\_\{4\}, letΓ1∈K4\\Gamma\_\{1\}\\in K\_\{4\}be the push\-map along the simple loop shown in Figure[6\.3](https://arxiv.org/html/2607.05283#S6.F3), and suppose thatΦ=Ψ⋅Γ1\\Phi=\\Psi\\cdot\\Gamma\_\{1\}is a proper product\. LetΔ\\Deltabe a disk arising in the disk sequence ofΦ\\Phi\. ThenΔ\\Deltais sign\-preserving if and only if𝒫​\(Δ\)\\mathcal\{P\}\(\\Delta\)is the following:

\{p2,p3\}\.\\hskip 7\.22743pt\\\{p\_\{2\},p\_\{3\}\\\}\.Moreover,Δ\\Deltais sign\-changing if and only if𝒫​\(Δ\)\\mathcal\{P\}\(\\Delta\)is one of the following:

\{p3\},\{p4\},\{p1,p2,p3\},\{p1,p2,p3,p4\}\.\\\{p\_\{3\}\\\},\\hskip 7\.22743pt\\\{p\_\{4\}\\\},\\hskip 7\.22743pt\\\{p\_\{1\},p\_\{2\},p\_\{3\}\\\},\\hskip 7\.22743pt\\\{p\_\{1\},p\_\{2\},p\_\{3\},p\_\{4\}\\\}\.

###### Proof\.

By Lemma[6\.2](https://arxiv.org/html/2607.05283#S6.Thmtheorem2)any diskΔ\\Deltain the disk sequence ofΦ\\Phican have itsβ\\beta\-component isotoped off ofΓ1\\Gamma\_\{1\}\. Referring to Figure[6\.4](https://arxiv.org/html/2607.05283#S6.F4), we see that, up to equivalence, there are only finitely many possibleβ\\beta\-subarcs forming anα\\alpha\-β\\betadisk that is disjoint fromΓ1\\Gamma\_\{1\}\. ∎

\\begin\{overpic\}\[width=195\.12767pt\]\{all\-disks\.png\} \\put\(10\.0,38\.0\)\{$p\_\{4\}$\} \\put\(37\.0,17\.0\)\{$p\_\{3\}$\} \\put\(74\.0,40\.0\)\{$p\_\{1\}$\} \\put\(53\.0,64\.0\)\{$p\_\{2\}$\} \\end\{overpic\}Figure 6\.4:The loopΓ1\\Gamma\_\{1\}is shown in gold, together with red arcs indicating all possible disks arising in the disk sequence of a braid inK4K\_\{4\}such that theβ\\beta\-component of the disk can be isotoped off ofΓ1\\Gamma\_\{1\}\.The five possible types of disk that can arise up to equivalence are shown in Figure[6\.5](https://arxiv.org/html/2607.05283#S6.F5)using our standard embedding ofα\\alphaandΓ1\\Gamma\_\{1\}inD4D\_\{4\}\.

\\begin\{overpic\}\[width=325\.215pt\]\{wn\-disks\.png\} \\put\(3\.0,69\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\delta\_\{1\}$\}\} \\put\(97\.0,64\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\delta\_\{2\}$\}\} \\put\(3\.0,41\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\delta\_\{3\}$\}\} \\put\(76\.0,29\.5\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\delta\_\{4\}$\}\} \\put\(8\.0,11\.0\)\{\{\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}$\\delta\_\{5\}$\}\} \\end\{overpic\}Figure 6\.5:The loopΓ1\\Gamma\_\{1\}is shown in gold, together with five red arcsδ1,…,δ5\\delta\_\{1\},\\ldots,\\delta\_\{5\}indicating all five types of disks arising in the disk sequence of a braid inK4K\_\{4\}such that theβ\\beta\-component of the disk can be isotoped off ofΓ1\\Gamma\_\{1\}\.

### 6\.3Embedding in𝐁5\{\\bm\{\\operatorname\{B\}\}\_\{5\}\}

Under the hypotheses of Lemma[6\.3](https://arxiv.org/html/2607.05283#S6.Thmtheorem3), there is only one possible type of disk arising in the disk sequence of a braid inK4K\_\{4\}that violates the parity condition: a disk containing all four punctures\. In order to deal with this, we will pass briefly to the setting of the disk with five punctures\. To that end, we now fix an embedding ofD4D\_\{4\}in the diskD5D\_\{5\}; we can think of this as simply fixing an additional marked pointp5p\_\{5\}in the interior of the diskD4D\_\{4\}\. IfΦ∈B4\\Phi\\in B\_\{4\}, we letf​\(Φ\)∈B5f\(\\Phi\)\\in B\_\{5\}denote the image ofΦ\\PhiinB5B\_\{5\}under the induced injectionf:B4↪B5f:\\operatorname\{B\}\_\{4\}\\hookrightarrow\\operatorname\{B\}\_\{5\}\. We emphasize that throughout this section, the notationΓ1\\Gamma\_\{1\}refers to the particular push\-map shown in Figure[6\.3](https://arxiv.org/html/2607.05283#S6.F3), whileΓ\\Gammarefers to a push\-map that generally depends on some other elementΦ∈K4\\Phi\\in K\_\{4\}\.

###### Proposition 6\.4\.

LetΦ∈B4\\Phi\\in\\operatorname\{B\}\_\{4\}, and suppose thatΦ\\Phican be written as a proper productΦ=Φ′⋅Γ1\\Phi=\\Phi^\{\\prime\}\\cdot\\Gamma\_\{1\}\. Then there exists a push\-mapΓ∈K5\\Gamma\\in K\_\{5\}such thatf​\(Φ\)⋅Γf\(\\Phi\)\\cdot\\Gammais a proper product inB5\\operatorname\{B\}\_\{5\}and such that bothf​\(Φ\)⋅Γf\(\\Phi\)\\cdot\\GammaandΓ\\Gammasatisfy the parity condition\. Furthermore we may chooseΓ\\Gammaso thatι​\(β∗3⋅\(Φ⋅Γ\),α\)≠ι​\(β∗3⋅Γ,α\)\\iota\(\\beta\_\{\*\}^\{3\}\\cdot\(\\Phi\\cdot\\Gamma\),\\alpha\)\\neq\\iota\(\\beta\_\{\*\}^\{3\}\\cdot\\Gamma,\\alpha\)\.

Before we begin the proof of Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4), we recall that anα\\alpha\-γ\\gammadisk is a disk whose boundary is the union of a single subarc ofα\\alphatogether with a single subarc ofγ\\gamma, whereγ\\gammais a loop or an arc\. In the case whereγ\\gammais a union of two or more subarcs of a larger loop or arcΓ\\Gamma, we will also refer to such a disk as anα​\-​Γ\\alpha\\mbox\{\-\}\\Gammadisk for convenience\.

###### Proof\.

Suppose that the braidΦ∈B4\\Phi\\in\\operatorname\{B\}\_\{4\}does not satisfy the parity condition\. By Lemma[6\.3](https://arxiv.org/html/2607.05283#S6.Thmtheorem3)there must be at least one diskΔ\\Deltain its disk sequence with𝒫​\(Δ\)=\{p1,p2,p3,p4\}\\mathcal\{P\}\(\\Delta\)=\\\{p\_\{1\},p\_\{2\},p\_\{3\},p\_\{4\}\\\}\. Moreover, any 4\-disks that occur in its disk sequence are necessarily nested inD4D\_\{4\}\. Consider the imagef​\(Φ\)f\(\\Phi\)of our braid inB5\\operatorname\{B\}\_\{5\}\. We will construct a push\-mapΓ∈K5\\Gamma\\in K\_\{5\}whose effect will be to replace each of the 4\-disks in the disk sequence off​\(Φ\)f\(\\Phi\)with a 5\-disk while preserving every other disk\.

Letβ\\betadenote the arc\(β∗3\)​f​\(Φ\)\(\\beta\_\{\*\}^\{3\}\)f\(\\Phi\)in the diskD5D\_\{5\}, and choose a pointqqin the interior of the innermost 4\-disk ofβ\\betasuch thatqqis not contained in akk\-disk in the disk sequence off​\(Φ\)f\(\\Phi\)for anyk≤3k\\leq 3\. Since the arcβ\\betais simple, its complement in the diskD5D\_\{5\}is connected, and hence we can choose a simple pathγ1\\gamma\_\{1\}fromp5p\_\{5\}toqqsuch thatι​\(γ1,β\)=0\\iota\(\\gamma\_\{1\},\\beta\)=0; see Figure[6\.6](https://arxiv.org/html/2607.05283#S6.F6)for an example\. We note thatγ1\\gamma\_\{1\}necessarily intersectsα\\alpha, sinceqqlies in the interior of the diskΔ\\Deltaandp5p\_\{5\}does not\.

\\begin\{overpic\}\[width=303\.53267pt\]\{gamma2\-1\.png\} \\put\(98\.0,28\.0\)\{$p\_\{5\}$\} \\put\(78\.0,4\.0\)\{$D\_\{4\}\\subseteq D\_\{5\}$\} \\put\(31\.0,15\.0\)\{\{\\color\[rgb\]\{0\.78515625,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.78515625,0,0\}$\\beta$\}\} \\put\(22\.0,39\.0\)\{\{\\color\[rgb\]\{0,0\.78515625,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.78515625,0\}$q$\}\} \\put\(92\.5,35\.0\)\{\{\\color\[rgb\]\{1,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,1\}$\\gamma\_\{2\}$\}\} \\put\(65\.0,27\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\gamma\_\{1\}$\}\} \\end\{overpic\}Figure 6\.6:An example of the construction of the loopΓ=γ1∪γ2\\Gamma=\\gamma\_\{1\}\\cup\\gamma\_\{2\}\.Next we choose a simple pathγ2\\gamma\_\{2\}fromqqtop5p\_\{5\}that is disjoint fromα\\alphaand not isotopic toγ1\\gamma\_\{1\}, such that the intersection ofγ2\\gamma\_\{2\}andβ\\betaconsists of precisely one point in theβ\\beta\-component of each 4\-disk in the disk sequence ofΦ\\Phi\. In other words,γ2\\gamma\_\{2\}travels transversely through theβ\\beta\-component of each 4\-disk and then passes out ofD4D\_\{4\}top5p\_\{5\}without intersectingβ\\betaagain, and in particular without intersecting any other disks in the disk sequence ofΦ\\Phi; again, see Figure[6\.6](https://arxiv.org/html/2607.05283#S6.F6)for an example\. Finally, we define the loopΓ∈K5\\Gamma\\in K\_\{5\}to be the product of pathsγ2∘γ1\\gamma\_\{2\}\\circ\\gamma\_\{1\}\.

Now,Γ\\Gammadoes not intersect anynn\-disks in the disk sequence ofΦ\\Phiforn<4n<4\. Moreover, given any 4\-disk in the disk sequence ofΦ\\Phi,Γ\\Gammaintersects theα\\alpha\-component of the disk’s boundary some nonzero number of times before finally “exiting” the disk at its single point of intersection with theβ\\beta\-component of the disk’s boundary and continuing on top5p\_\{5\}\. This implies that the effect on the 4\-disk of pushingp5p\_\{5\}alongΓ\\Gammawill be to replace it with a 5\-disk containing\{p1,p2,p3,p4,p5\}\\\{p\_\{1\},p\_\{2\},p\_\{3\},p\_\{4\},p\_\{5\}\\\}\. As we have already observed, any 4\-disk in the disk sequence ofΦ\\Phimust be sign\-changing, and hence the newly created 5\-disk replacing is also necessarily sign\-changing\.

To summarize, the disk sequence of\(β\)​Γ=\(β∗3\)​\(Φ⋅Γ\)\(\\beta\)\\Gamma=\(\\beta\_\{\*\}^\{3\}\)\(\\Phi\\cdot\\Gamma\)is exactly the same as that ofβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phi, except that any 4\-disks in the original disk sequence ofΦ∈B4\\Phi\\in\\operatorname\{B\}\_\{4\}have now been replaced by a 5\-disk\. Hencef​\(Φ\)⋅Γ∈B5f\(\\Phi\)\\cdot\\Gamma\\in\\operatorname\{B\}\_\{5\}satisfies the parity condition\.

Consider next the disk sequence of\(β∗3\)⋅Γ\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma\. We claim that any disk in this sequence is either a 5\-disk, a 1\-disk containingp5p\_\{5\}, or is equivalent to a disk formed withδi\\delta\_\{i\}for somei≥2i\\geq 2as shown in Figure[6\.5](https://arxiv.org/html/2607.05283#S6.F5); it then follows thatΓ\\Gammaalso satisfies the parity condition\.

To see this, we first note that by constructionγ1\\gamma\_\{1\}is a simple arc, and we may assume that it admits no bigon\-forming polygons withβ∗3\\beta\_\{\*\}^\{3\}andα\\alpha\. Sinceγ1\\gamma\_\{1\}is disjoint fromβ=\(β∗3\)​\(Φ\)\\beta=\(\\beta\_\{\*\}^\{3\}\)\(\\Phi\), and sinceΦ\\Phican be written as a proper product of the formΦ′⋅Γ1\\Phi^\{\\prime\}\\cdot\\Gamma\_\{1\}, we may further assume without loss of generality that eachα\\alpha\-γ1\\gamma\_\{1\}disk is equivalent to one of the five disks shown in Figure[6\.5](https://arxiv.org/html/2607.05283#S6.F5); otherwise we could have made a different choice ofγ1\\gamma\_\{1\}\. It follows that anyα\\alpha\-β\\betadisks that result from pushingβ∗3\\beta\_\{\*\}^\{3\}alongγ1\\gamma\_\{1\}will also be equivalent to one of the five disks shown in Figure[6\.5](https://arxiv.org/html/2607.05283#S6.F5), or else it is a 1\-disk containing onlyp5p\_\{5\}\. In order to complete the push aroundΓ\\Gamma, we next push alongγ2\\gamma\_\{2\}\. As noted above, the effect of doing so will be to replace any existing 4\-disks formed in the first stage with 5\-disks\. The claim follows, and we have established the first statement of the proposition\.

It remains to show that we can always chooseΓ\\Gammato ensure the following statement holds:

ι​\(α,\(β∗3\)⋅Γ\)≠ι​\(α,\(β∗3\)​\(Φ⋅Γ\)\)\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma\)\\neq\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\(\\Phi\\cdot\\Gamma\)\)\(2\)
Suppose now that in our construction ofΓ\\Gammaas above, we haveι​\(α,\(β∗3\)⋅Γ\)=ι​\(α,\(β∗3\)​\(Φ⋅Γ\)\)\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma\)=\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\(\\Phi\\cdot\\Gamma\)\), or in other words,ι​\(α,\(β∗3\)​Γ\)=ι​\(α,\(β\)​Γ\)\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\Gamma\)=\\iota\(\\alpha,\(\\beta\)\\Gamma\)\. Our strategy is to alter our construction of the loopΓ\\Gammaslightly so that the left\-hand side of this equality increases, while the right\-hand side remains unchanged\.

To this end, we will form a new loopΓ′\\Gamma^\{\\prime\}starting with the same choice ofγ1\\gamma\_\{1\}as before, but we will choose a different arcγ2′\\gamma\_\{2\}^\{\\prime\}joining the pointqqtop5p\_\{5\}to replaceγ2\\gamma\_\{2\}, and setΓ′=γ1∪γ2′\\Gamma^\{\\prime\}=\\gamma\_\{1\}\\cup\\gamma\_\{2\}^\{\\prime\}\. We note that it is sufficient to find such an arcγ2′\\gamma\_\{2\}^\{\\prime\}satisfying the following two properties:

1. 1\.as we traverseγ2′\\gamma\_\{2\}^\{\\prime\}fromqqtop5p\_\{5\}, its first point of intersection withβ=\(β∗3\)⋅\(Φ\)\\beta=\(\\beta\_\{\*\}^\{3\}\)\\cdot\(\\Phi\)does not occur until after its final point of intersection withα\\alpha; and
2. 2\.ι​\(α,\(β∗3\)⋅Γ′\)\>ι​\(α,\(β∗3\)⋅Γ\)\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma^\{\\prime\}\)\>\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gamma\)\.

This follows from the fact that the first condition implies thatι​\(α,\(β\)​Γ′\)=ι​\(α,\(β\)​Γ\)\\iota\(\\alpha,\(\\beta\)\\Gamma^\{\\prime\}\)=\\iota\(\\alpha,\(\\beta\)\\Gamma\)\.

Revisiting the construction ofΓ\\Gamma, letssdenote the final point of intersection ofγ1\\gamma\_\{1\}withα\\alphaas we travel alongγ1\\gamma\_\{1\}, and let\(γ1\)F\(\\gamma\_\{1\}\)\_\{F\}denote the subarc ofγ1\\gamma\_\{1\}joiningssto the pointqq\. SinceD4\\\(β∪γ1\)D\_\{4\}\\backslash\(\\beta\\cup\\gamma\_\{1\}\)is connected, there exists an arcγ′\\gamma^\{\\prime\}joiningqqto a pointr∈αr\\in\\alpha, where the pointrrdepends on the choice ofγ′\\gamma^\{\\prime\}, such thatγ′\\gamma^\{\\prime\}is disjoint fromβ\\betaand such thatγ′∩γ1=\{q\}\\gamma^\{\\prime\}\\cap\\gamma\_\{1\}=\\\{q\\\}\.

We will now consider two cases\. First, suppose thatγ′\\gamma^\{\\prime\}can be chosen so that, the arcγ′∪\(γ1\)F\\gamma^\{\\prime\}\\cup\(\\gamma\_\{1\}\)\_\{F\}, together with the subarc ofα\\alphajoiningrrtoss, forms a diskΔ\\Deltacontainingp1,p2,p3p\_\{1\},p\_\{2\},p\_\{3\}, andp4p\_\{4\}\.

\\begin\{overpic\}\[width=303\.53267pt\]\{gamma2\-const\.png\} \\put\(98\.0,28\.0\)\{$p\_\{5\}$\} \\put\(78\.0,4\.0\)\{$D\_\{4\}\\subseteq D\_\{5\}$\} \\put\(29\.0,38\.0\)\{\{\\color\[rgb\]\{0,0\.78515625,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.78515625,0\}$q$\}\} \\put\(36\.0,34\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$s$\}\} \\put\(21\.0,34\.0\)\{\{\\color\[rgb\]\{0,0\.78515625,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.78515625,0\}$r$\}\} \\put\(22\.0,37\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\gamma^\{\\prime\\prime\}$\}\} \\put\(14\.0,30\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\gamma^\{\\prime\}$\}\} \\put\(60\.0,49\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\\gamma\_\{2\}$\}\} \\put\(65\.0,27\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}$\(\\gamma\_\{1\}\)\_\{F\}$\}\} \\end\{overpic\}Figure 6\.7:An alternate construction ofγ2\\gamma\_\{2\}: Case 1\.We next look for an arcγ′′\\gamma^\{\\prime\\prime\}fromrrback toqqsatisfying the following:

1. 1\.the arcγ′′\\gamma^\{\\prime\\prime\}is disjoint fromβ\\beta;
2. 2\.γ′′∩\(γ1∪γ′\)=\{q,r\}\\gamma^\{\\prime\\prime\}\\cap\(\\gamma\_\{1\}\\cup\\gamma^\{\\prime\}\)=\\\{q,r\\\}; and
3. 3\.γ′′\\gamma^\{\\prime\\prime\}is not isotopic toγ′\\gamma^\{\\prime\}relative to the endpoints\{q,r\}\\\{q,r\\\}\.

This is equivalent to finding a nontrivial loop based atrrinD4\\\(β∪γ1∪γ′\)D\_\{4\}\\backslash\(\\beta\\cup\\gamma\_\{1\}\\cup\\gamma^\{\\prime\}\)\. This is always possible sinceβ∩\(γ1∪γ′\)=∅\\beta\\cap\(\\gamma\_\{1\}\\cup\\gamma^\{\\prime\}\)=\\emptyset, and henceD4\\\(β∪γ1∪γ′\)D\_\{4\}\\backslash\(\\beta\\cup\\gamma\_\{1\}\\cup\\gamma^\{\\prime\}\)is homeomorphic to a disk with three punctures:p1,p2,p\_\{1\},p\_\{2\},andp4p\_\{4\}\. Finally, we setγ2′=γ′∪γ′′∪γ2\\gamma\_\{2\}^\{\\prime\}=\\gamma^\{\\prime\}\\cup\\gamma^\{\\prime\\prime\}\\cup\\gamma\_\{2\}, and setΓ′=γ1∪γ2′\\Gamma^\{\\prime\}=\\gamma\_\{1\}\\cup\\gamma\_\{2\}^\{\\prime\}\. Since the subarcγ2′\\gamma\_\{2\}^\{\\prime\}satisfies the two properties above, we have established the result in this case\.

It remains to deal with the case where the arcγ′\\gamma^\{\\prime\}joiningqqto a pointr∈αr\\in\\alphacannot be chosen so thatγ′∪\(γ1\)F\\gamma^\{\\prime\}\\cup\(\\gamma\_\{1\}\)\_\{F\}forms a 4\-disk, and hence forms a disk containing a proper subset of\{p1,p2,p3,p4\}\\\{p\_\{1\},p\_\{2\},p\_\{3\},p\_\{4\}\\\}; recall that we are still using in this case thatγ′\\gamma^\{\\prime\}is disjoint fromβ\\betaand thatγ′∩γ1=\{q\}\\gamma^\{\\prime\}\\cap\\gamma\_\{1\}=\\\{q\\\}\.

We will choose the arcγ′\\gamma^\{\\prime\}to travel fromqqto a pointr∈αr\\in\\alphain parallel to theβ\\beta\-component of the boundary of the innermost 4\-disk; there are two possibilities here\. One choice will result in\(γ1\)F∪γ′\(\\gamma\_\{1\}\)\_\{F\}\\cup\\gamma^\{\\prime\}forming anα​\-​Γ\\alpha\\mbox\{\-\}\\Gammadisk containingkkpunctures, wherek<4k<4, and the other choice forγ′\\gamma^\{\\prime\}results in anα​\-​Γ\\alpha\\mbox\{\-\}\\Gammadisk containingn−kn\-kpunctures\. Referring to Figure[6\.8](https://arxiv.org/html/2607.05283#S6.F8), we see that hereγ′\\gamma^\{\\prime\}is chosen to follow the innermost suchβ\\beta\-component in the right\-hand direction \(as shown in the figure\), so that\(γ1\)F∪γ′\(\\gamma\_\{1\}\)\_\{F\}\\cup\\gamma^\{\\prime\}forms a 1\-disk withα\\alpha; following to the left would form a 3\-disk instead\. In other words, we can assume without loss of generality that theα​\-​Γ\\alpha\\mbox\{\-\}\\Gammadisk formed by\(γ1\)F∪γ′\(\\gamma\_\{1\}\)\_\{F\}\\cup\\gamma^\{\\prime\}contains at most two punctures\.

\\begin\{overpic\}\[width=260\.17464pt\]\{case2\-partitions\.png\} \\put\(40\.0,40\.0\)\{\{\\color\[rgb\]\{0,0\.78515625,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.78515625,0\}\\small$q$\}\} \\put\(41\.0,56\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}\\small$\\gamma^\{\\prime\}$\}\} \\put\(50\.0,22\.0\)\{\{\\color\[rgb\]\{1,0\.6953125,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0\.6953125,0\}\\small$\(\\gamma\_\{1\}\)\_\{F\}$\}\} \\end\{overpic\}Figure 6\.8:The horizontal lines in this figure representβ\\beta\-components of 4\-disks\. The choice of the arcγ′\\gamma^\{\\prime\}shown here yields a disk with 1 puncture, by traveling to the right in parallel with theβ\\beta\-component of the innermost 4\-disk\.Our construction now reduces as in the previous case to finding a nontrivial loop based atrrinDn\\\(β∪γ1∪γ′∪γ′′\)D\_\{n\}\\backslash\(\\beta\\cup\\gamma\_\{1\}\\cup\\gamma^\{\\prime\}\\cup\\gamma^\{\\prime\\prime\}\)\. Again, this is always possible sinceβ∩\(γ1∪γ′\)=∅\\beta\\cap\(\\gamma\_\{1\}\\cup\\gamma^\{\\prime\}\)=\\emptyset, and since by constructionD4\\\(β∪γ1∪γ′\)D\_\{4\}\\backslash\(\\beta\\cup\\gamma\_\{1\}\\cup\\gamma^\{\\prime\}\)is homeomorphic to a disk with at least one puncture\. This completes the proof of the proposition\. ∎

We remark that the same construction used to find the loopΓ\\Gammain the proof of Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4)could be iterated to find, for anynn, and for anyΦ∈Bn\\Phi\\in B\_\{n\}, someΓ1,…,Γk\\Gamma\_\{1\},\\dots,\\Gamma\_\{k\}so that, for the appropriate inclusion mapff, we have thatf​\(Φ\)⋅Γ1​…​Γk∈Bn\+kf\(\\Phi\)\\cdot\\Gamma\_\{1\}\\dots\\Gamma\_\{k\}\\in B\_\{n\+k\}satisfies the parity condition\. The rough idea here is to start with the innermost disk of each occuring type that does not satisfy the parity condition, and find a suitable push\-map that “corrects” its parity\. However, in this case we cannot guarantee that the braidΓ1​…​Γk\\Gamma\_\{1\}\\dots\\Gamma\_\{k\}satisfies the parity condition\. In our case, the fact the braidΓ\\Gammaalso satisfies the parity condition is important in the proof of our Main Theorem\. The fact that we only need to correct one type of disk, so we only require one push mapΓ\\Gamma, is crucial\. In the general case wherek\>1k\>1, we would not be able to simultaneously assume thatΓ2\\Gamma\_\{2\}does not cross bothβ\\betaand\(β\)⋅Γ1\(\\beta\)\\cdot\\Gamma\_\{1\}, and so this argument would fail\.

The following corollary of Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4)enables us to derive conditions on the Moody polynomial of a push\-mapΦ∈K4\\Phi\\in K\_\{4\}, as part of a proper product\.

###### Corollary 6\.5\.

IfΦ∈K4\\Phi\\in K\_\{4\}admits a proper productΦ=Φ′⋅Γ1\\Phi=\\Phi^\{\\prime\}\\cdot\\Gamma\_\{1\}and satisfiesι​\(α,\(β∗3\)​Φ\)\>0\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\Phi\)\>0, then there exists a push\-mapΓ∈K5\\Gamma\\in K\_\{5\}such that𝕄f​\(Φ\)⋅Γ≠𝕄Γ\\mathbb\{M\}\_\{f\(\\Phi\)\\cdot\\Gamma\}\\neq\\mathbb\{M\}\_\{\\Gamma\}\.

###### Proof\.

LetΦ\\Phibe a braid inK4K\_\{4\}, and letΓ\\Gammabe an associated push\-map inK5K\_\{5\}as in the statement of Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4)\. By Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4), bothΓ\\GammaandΦ⋅Γ\\Phi\\cdot\\Gammasatisfy the parity condition\. It then follows from Lemma[4\.2](https://arxiv.org/html/2607.05283#S4.Thmtheorem2)that there are no cancellations in either of the corresponding Moody polynomials𝕄Γ\\mathbb\{M\}\_\{\\Gamma\}and𝕄f​\(Φ\)⋅Γ\\mathbb\{M\}\_\{f\(\\Phi\)\\cdot\\Gamma\}\.

LetM−M\_\{\-\}andM\+M\_\{\+\}be the lowest and highest exponents that appear in the Moody polynomial𝕄f​\(Φ\)⋅Γ\\mathbb\{M\}\_\{f\(\\Phi\)\\cdot\\Gamma\}, and similarly letM−′M\_\{\-\}^\{\\prime\}andM\+′M\_\{\+\}^\{\\prime\}denote the corresponding values for𝕄Γ\\mathbb\{M\}\_\{\\Gamma\}\. Then, since neither polynomial has any cancellations, we can write the corresponding Moody polynomials as follows:

𝕄f​\(Φ\)⋅Γ=∑j=M−M\+aj​tj\\displaystyle\\mathbb\{M\}\_\{f\(\\Phi\)\\cdot\\Gamma\}=\\sum\_\{j=M\_\{\-\}\}^\{M\_\{\+\}\}a\_\{j\}t^\{j\}=\\displaystyle=\|aM−\|\+⋯\+\|aM\+\|=ι​\(α,\(β∗3\)​\(f​\(Φ\)⋅Γ\)\);and\\displaystyle\|a\_\{M\_\{\-\}\}\|\+\\dots\+\|a\_\{M\_\{\+\}\}\|=\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\(f\(\\Phi\)\\cdot\\Gamma\)\);\\qquad\\mbox\{ and \}𝕄Γ=∑j=M−′M\+′bj​tj\\displaystyle\\mathbb\{M\}\_\{\\Gamma\}=\\sum\_\{j=M^\{\\prime\}\_\{\-\}\}^\{M^\{\\prime\}\_\{\+\}\}b\_\{j\}t^\{j\}=\\displaystyle=\|bM−′\|\+⋯\+\|bM\+′\|=ι​\(α,\(β∗3\)​Γ\)\.\\displaystyle\|b\_\{M\_\{\-\}^\{\\prime\}\}\|\+\\dots\+\|b\_\{M^\{\\prime\}\_\{\+\}\}\|=\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\Gamma\)\.By Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4)we can assume thatι​\(α,\(β∗3\)​\(f​\(Φ\)⋅Γ\)\)≠ι​\(α,\(β∗3\)​Γ\)\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\(f\(\\Phi\)\\cdot\\Gamma\)\)\\neq\\iota\(\\alpha,\(\\beta\_\{\*\}^\{3\}\)\\Gamma\)\. The result follows\. ∎

### 6\.4Faithfulness on𝐁𝐫𝐮𝐧4\{\\bm\{\\operatorname\{Brun\}\}\_\{4\}\}

As described in Section[1](https://arxiv.org/html/2607.05283#S1), the proof of our Main Theorem follows from the next result\.

###### Theorem 6\.6\.

The Burau representationρ4\\rho\_\{4\}is faithful on its restriction toBrun4\\operatorname\{Brun\}\_\{4\}\.

###### Proof\.

LetΦ\\Phidenote a nontrivial element inBrun4\\operatorname\{Brun\}\_\{4\}\. We make the following elementary observation:Φ∈Brun4\\Phi\\in\\operatorname\{Brun\}\_\{4\}lies in the kernel of the Burau representationρ4\\rho\_\{4\}if and only if, for every elementy∈B4y\\in\\operatorname\{B\}\_\{4\}, the conjugatey​Φ​y−1y\\Phi y^\{\-1\}also lies in the kernel ofρ4\\rho\_\{4\}\. We also recall from our discussion in Section[1](https://arxiv.org/html/2607.05283#S1)that any nontrivial element ofBrun4\\operatorname\{Brun\}\_\{4\}is pseudo\-Anosov, and that the image of the arcβ∗3\\beta\_\{\*\}^\{3\}under a pseudo\-Anosov map necessarily intersects the arcα\\alphanontrivially\.

Given a simple loop such asΓ1\\Gamma\_\{1\}the only possible bigon\-forming polygons that can occur areβ\\beta\-to\-α\\alphatrigons and rectangles\. Now,β\\beta\-to\-α\\alphatrigons can be resolved by an isotopy ofβ\\beta, and rectangles are not possible in this case since the points of intersection betweenα\\alphaandΓ1\\Gamma\_\{1\}alternate in sign as we travel alongα\\alpha\.

Furthermore, if we considerΦ\\PhiandΓ1\\Gamma\_\{1\}as freely reduced words in the free groupK4K\_\{4\}with respect to the free basis shown in Figure[6\.9](https://arxiv.org/html/2607.05283#S6.F9), and if the juxtaposition of the two words \(yielding the productΦ⋅Γ1\\Phi\\cdot\\Gamma\_\{1\}\) is also a freely reduced word with respect to this free basis, then the second condition forΦ⋅Γ1\\Phi\\cdot\\Gamma\_\{1\}to be a proper product is immediately satisfied\. One way to see this is to note that, as a freely reduced word in this basis,Γ1\\Gamma\_\{1\}begins withy2−1y\_\{2\}^\{\-1\}\. IfΦ⋅Γ1\\Phi\\cdot\\Gamma\_\{1\}is freely reduced, thenΦ\\Phicannot end withy2y\_\{2\}; if it does, we can replaceΦ\\Phiwith an appropriate conjugate that does not end withy2y\_\{2\}\. Thus we may assume thatΦ\\Phihas been conjugated so that it can be written as a proper productΦ=Φ′⋅Γ1\\Phi=\\Phi^\{\\prime\}\\cdot\\Gamma\_\{1\}for someΦ′∈K4\\Phi^\{\\prime\}\\in K\_\{4\}\.

\\begin\{overpic\}\[width=130\.08731pt\]\{free\-basis\.png\} \\put\(54\.0,33\.0\)\{$y\_\{1\}$\} \\put\(22\.0,33\.0\)\{$y\_\{2\}$\} \\put\(\-6\.5,33\.0\)\{$y\_\{3\}$\} \\end\{overpic\}Figure 6\.9:A free basis forK4K\_\{4\}\.By Corollary[6\.5](https://arxiv.org/html/2607.05283#S6.Thmtheorem5), there is some push\-mapΓ∈K5<B5\\Gamma\\in K\_\{5\}<\\operatorname\{B\}\_\{5\}so that𝕄f​\(Φ\)⋅Γ≠𝕄Γ\\mathbb\{M\}\_\{f\(\\Phi\)\\cdot\\Gamma\}\\neq\\mathbb\{M\}\_\{\\Gamma\}\. It now follows from Theorem[2\.3](https://arxiv.org/html/2607.05283#S2.Thmtheorem3)thatf​\(Φ\)f\(\\Phi\)does not lie in the kernel ofρ5\\rho\_\{5\}, and henceΦ\\Phidoes not lie in the kernel ofρ4\\rho\_\{4\}\. Hence the Burau representation ofB4B\_\{4\}is faithful on the point\-pushing subgroupBrun4\\operatorname\{Brun\}\_\{4\}\. ∎

## 7An example

We end with an illustrative example of a push\-mapΓ∈K5\\Gamma\\in K\_\{5\}of the form guaranteed by Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4)\. We will applyΓ\\Gammato an arcβ\\betawhose corresponding Moody polynomial admits cancellation, and we will see that after applyingΓ\\Gammathere is no longer any cancellation\. Letβ\\betadenote the arc shown in Figure[7\.1](https://arxiv.org/html/2607.05283#S7.F1)\. We recall that an arcβ\\betadoes not uniquely determine a braidΦ\\Phisuch thatβ=\(β∗3\)​Φ\\beta=\(\\beta\_\{\*\}^\{3\}\)\\Phi; nevertheless in what follows it will be useful to make a choice of such aΦ\\Phito simplify notation\.

To simplify the calculation while explaining the important parts of the construction, we have chosen to specify an arcβ\\betawhose disk sequence satisfies the same properties as those guaranteed by Lemma[6\.3](https://arxiv.org/html/2607.05283#S6.Thmtheorem3), but our choice ofβ\\betadoes not arise as\(β∗3\)​Φ\(\\beta\_\{\*\}^\{3\}\)\\Phifor any choice ofΦ∈K4\\Phi\\in K\_\{4\}\. \(Recall that we restricted toK4K\_\{4\}in Theorem[6\.6](https://arxiv.org/html/2607.05283#S6.Thmtheorem6)in order to ensure thatΦ\\Phican be written as a proper productΦ=Φ′⋅Γ1\\Phi=\\Phi^\{\\prime\}\\cdot\\Gamma\_\{1\}, which allows us to apply Lemma[6\.3](https://arxiv.org/html/2607.05283#S6.Thmtheorem3)\.\) We make this choice for illustrative purposes because the simplest example of an arcβ\\betaadmitting cancellations, and arising from a proper product of the formΦ=Φ′⋅Γ1∈K4\\Phi=\\Phi^\{\\prime\}\\cdot\\Gamma\_\{1\}\\in K\_\{4\}, would have many more intersections\.

![Refer to caption](https://arxiv.org/html/2607.05283v1/beta-ex.png)Figure 7\.1:An example of an arcβ\\betawhose Moody polynomial admits a cancellation\.Note that any braidΦ\\Phiinducing the arcβ\\betadoes not satisfy the parity condition\. Here and throughout this appendix, we write the terms of the unsimplified Moody polynomial in order corresponding to points of intersection ofα\\alphawithβ\\betaas we travel alongβ\\betafrom the basepoint top3p\_\{3\}\. We compute the Moody polynomial forΦ\\Phias follows, noting that the linear terms appear with opposite signs, yielding a cancellation\.

𝕄Φ\\displaystyle\\mathbb\{M\}\_\{\\Phi\}=\\displaystyle=t0\+t2\+t4−t3−t\+t−3\+t−1\+t\\displaystyle t^\{0\}\+t^\{2\}\+t^\{4\}\-t^\{3\}\-t\+t^\{\-3\}\+t^\{\-1\}\+t=\\displaystyle=1\+t2\+t4−t3\+t−3\+t−1\.\\displaystyle 1\+t^\{2\}\+t^\{4\}\-t^\{3\}\+t^\{\-3\}\+t^\{\-1\}\.
Following the procedure given in the proof of Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4), we obtain the loopΓ∈K5\\Gamma\\in K\_\{5\}shown in Figure[7\.2](https://arxiv.org/html/2607.05283#S7.F2)\.

\\begin\{overpic\}\[width=303\.53267pt\]\{gamma\-b5\.png\} \\put\(74\.0,15\.0\)\{\\small$q$\} \\end\{overpic\}Figure 7\.2:The push mapΓ∈K5\\Gamma\\in K\_\{5\}shown in gold, with the pointqqmarked from the proof of Proposition[6\.4](https://arxiv.org/html/2607.05283#S6.Thmtheorem4)\.Pushingβ\\betaalongΓ\\Gammagives usβ⋅Γ\\beta\\cdot\\Gammaas shown in Figure[7\.3](https://arxiv.org/html/2607.05283#S7.F3), which now satisfies the parity condition\.

![Refer to caption](https://arxiv.org/html/2607.05283v1/b5-ex.png)Figure 7\.3:The image\(β\)​Γ\(\\beta\)\\GammainD5D\_\{5\}We next compute the Moody polynomial of the arcΦ⋅Γ\\Phi\\cdot\\Gamma:

𝕄Φ⋅Γ​\(t\)\\displaystyle\\mathbb\{M\}\_\{\\Phi\\cdot\\Gamma\}\(t\)=\\displaystyle=t0\+t2\+t4−t3−t\+t−4\+t−2\+t0\\displaystyle t^\{0\}\+t^\{2\}\+t^\{4\}\-t^\{3\}\-t\+t^\{\-4\}\+t^\{\-2\}\+t^\{0\}=\\displaystyle=2\+t2\+t4−t3−t\+t−4\+t−2\.\\displaystyle 2\+t^\{2\}\+t^\{4\}\-t^\{3\}\-t\+t^\{\-4\}\+t^\{\-2\}\.Comparing with our calculation of the unsimplified Moody polynomial ofΦ\\Phiabove, we see that the effect ofΓ\\Gammaon the Moody polynomial has been to increase the exponent of each the last three terms \(those following the appearance of the 5\-disk\) by 1, and there is no longer any cancellation\.

Furthermore, the arc\(β∗3\)⋅Γ\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gammadetermining the Moody polynomial ofΓ\\Gammais shown in Figure[7\.4](https://arxiv.org/html/2607.05283#S7.F4)and satisfies the parity condition as well\. The unsimplified Moody polynomial for the arc\(β∗3\)⋅Γ\(\\beta\_\{\*\}^\{3\}\)\\cdot\\Gammahas 20 terms; the key point here is that anyβ\\beta\-arc satisfying the parity condition may have any number of terms appearing with equal exponents, but every term with the same exponent will have the same sign\.

𝕄Γ​\(t\)\\displaystyle\\mathbb\{M\}\_\{\\Gamma\}\(t\)=\\displaystyle=−t0−t2\+t\+t−1−t0−t2\+t3\+t\+t−1−t−2−t0\+t−1\+t−3−t−2−t0\+t\+t−1\+t−3−t−2−t0\\displaystyle\-t^\{0\}\-t^\{2\}\+t\+t^\{\-1\}\-t^\{0\}\-t^\{2\}\+t^\{3\}\+t\+t^\{\-1\}\-t^\{\-2\}\-t^\{0\}\+t^\{\-1\}\+t^\{\-3\}\-t^\{\-2\}\-t^\{0\}\+t\+t^\{\-1\}\+t^\{\-3\}\-t^\{\-2\}\-t^\{0\}=\\displaystyle=2​t−3−3​t−2\+4​t−1−5\+3​t−2​t2\+t3\.\\displaystyle 2t^\{\-3\}\-3t^\{\-2\}\+4t^\{\-1\}\-5\+3t\-2t^\{2\}\+t^\{3\}\.
![Refer to caption](https://arxiv.org/html/2607.05283v1/gamma-im-ex.png)Figure 7\.4:The image\(β∗3\)⋅Γ\(\\beta\_\{\*\}^\{3\}\)\\cdot\\GammainD5D\_\{5\}\.
## References

- \[AFN02\]R\. C\. Alperin, B\. Farb, and G\. A\. Noskov\(2002\-07\)A Strong Schottky Lemma for Nonpositively Curved Singular Spaces\.Geometriae Dedicata92\(1\),pp\. 235–243\.External Links:ISSN 1572\-9168,[Link](https://doi.org/10.1023/A:1019650112871),[Document](https://dx.doi.org/10.1023/A%3A1019650112871)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[BT18a\]A\. Beridze and P\. Traczyk\(2018\)Burau representation forn=4n=4\.J\. Knot Theory Ramifications27\(3\),pp\. 1840002, 6\.External Links:ISSN 0218\-2165,1793\-6527,[Document](https://dx.doi.org/10.1142/S0218216518400023),[Link](https://doi.org/10.1142/S0218216518400023),[MathReview \(Eric C\. Rowell\)](https://www.ams.org/mathscinet-getitem?mr=3786776)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[BT18b\]A\. Beridze and P\. Traczyk\(2018\)Forks, noodles and the Burau representation forn=4n=4\.Trans\. A\. Razmadze Math\. Inst\.172\(3\),pp\. 337–353\.External Links:ISSN 2346\-8092,2588\-9028,[Document](https://dx.doi.org/10.1016/j.trmi.2018.05.001),[Link](https://doi.org/10.1016/j.trmi.2018.05.001),[MathReview \(Dimos Goundaroulis\)](https://www.ams.org/mathscinet-getitem?mr=3870051)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[BIG99\]S\. Bigelow\(1999\)The Burau representation is not faithful forn=5n=5\.Geom\. Topol\.3,pp\. 397–404\.External Links:[Document](https://dx.doi.org/10.2140/gt.1999.3.397),ISSN 1465\-3060,[Link](https://doi-org.ezproxy.lib.gla.ac.uk/10.2140/gt.1999.3.397),[MathReview \(Darren D\. Long\)](https://www.ams.org/mathscinet-getitem?mr=1725480)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p2.5),[§1](https://arxiv.org/html/2607.05283#S1.p5.4),[§2](https://arxiv.org/html/2607.05283#S2.p11.23),[§2](https://arxiv.org/html/2607.05283#S2.p5.2),[§2](https://arxiv.org/html/2607.05283#S2.p7.18),[Lemma 3\.1](https://arxiv.org/html/2607.05283#S3.Thmtheorem1),[§3](https://arxiv.org/html/2607.05283#S3.p4.4)\.
- \[BB05\]J\. S\. Birman and T\. E\. Brendle\(2005\)Braids: a survey\.InHandbook of knot theory,pp\. 19–103\.External Links:[Document](https://dx.doi.org/10.1016/B978-044451452-3/50003-4),[Link](https://doi-org.ezproxy.lib.gla.ac.uk/10.1016/B978-044451452-3/50003-4),[MathReview \(Darren D\. Long\)](https://www.ams.org/mathscinet-getitem?mr=2179260)Cited by:[§2](https://arxiv.org/html/2607.05283#S2.p1.12)\.
- \[BIR74\]J\. S\. Birman\(1974\)Braids, links, and mapping class groups\.Princeton University Press, Princeton, N\.J\.; University of Tokyo Press, Tokyo\.Note:Annals of Mathematics Studies, No\. 82External Links:[MathReview \(Wilbur Whitten\)](https://www.ams.org/mathscinet-getitem?mr=0375281)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p5.4)\.
- \[BUR35\]W\. Burau\(1935\)Über Zopfgruppen und gleichsinnig verdrillte Verkettungen\.Abh\. Math\. Sem\. Univ\. Hamburg11\(1\),pp\. 179–186\.External Links:[Document](https://dx.doi.org/10.1007/BF02940722),ISSN 0025\-5858,[Link](https://doi-org.ezproxy.lib.gla.ac.uk/10.1007/BF02940722),[MathReview Entry](https://www.ams.org/mathscinet-getitem?mr=3069652)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p1.9)\.
- \[CI17\]M\. Calvez and T\. Ito\(2017\)A Garside\-theoretic analysis of the Burau representations\.J\. Knot Theory Ramifications26\(7\),pp\. 1750040, 29\.External Links:ISSN 0218\-2165,1793\-6527,[Document](https://dx.doi.org/10.1142/S0218216517500407),[Link](https://doi.org/10.1142/S0218216517500407),[MathReview \(Jean Fromentin\)](https://www.ams.org/mathscinet-getitem?mr=3660095)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[DAT22\]A\. Datta\(2022\)A strong characterization of the entries of the Burau matrices of 4\-braids: The Burau representation of the braid groupB4B\_\{4\}is faithful almost everywhere\.Note:PreprintCited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[DLU24\]E\. Dlugie\(2024\)The Burau representation and shapes of polyhedra\.Algebr\. Geom\. Topol\.24\(5\),pp\. 2787–2805\.External Links:ISSN 1472\-2747,1472\-2739,[Document](https://dx.doi.org/10.2140/agt.2024.24.2787),[Link](https://doi-org.ezproxy2.lib.gla.ac.uk/10.2140/agt.2024.24.2787),[MathReview \(Mohammad N\. Abdulrahim\)](https://www.ams.org/mathscinet-getitem?mr=4790753)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[FM12\]B\. Farb and D\. Margalit\(2012\)A primer on mapping class groups\.Princeton Mathematical Series, Vol\.49,Princeton University Press, Princeton, NJ\.External Links:ISBN 978\-0\-691\-14794\-9,[MathReview \(Stephen P\. Humphries\)](https://www.ams.org/mathscinet-getitem?mr=2850125)Cited by:[§5](https://arxiv.org/html/2607.05283#S5.p2.16)\.
- \[FS19\]N\. J\. Fullarton and R\. Shadrach\(2019\)Observed periodicity related to the four\-strand Burau representation\.Exp\. Math\.28\(1\),pp\. 57–66\.External Links:ISSN 1058\-6458,1944\-950X,[Document](https://dx.doi.org/10.1080/10586458.2017.1335625),[Link](https://doi-org.ezproxy1.lib.gla.ac.uk/10.1080/10586458.2017.1335625),[MathReview \(Stefan Witzel\)](https://www.ams.org/mathscinet-getitem?mr=3938578)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[GWY25\]J\. Gibson, G\. Williamson, and O\. Yacobi\(2025\)4\-strand Burau is unfaithful modulo 5\.J\. Exp\. Math\.1\(1\),pp\. 36–50\.External Links:[Document](https://dx.doi.org/),ISSN,LinkCited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[JON87\]V\. F\. R\. Jones\(1987\)Hecke algebra representations of braid groups and link polynomials\.Annals of Mathematics126\(2\),pp\. 335–388\.External Links:ISSN 0003486X, 19398980,[Link](http://www.jstor.org/stable/1971403)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.SS0.SSS0.Px1.p1.3)\.
- \[KAS08\]Y\. Kasahara\(2008\)Remarks on the faithfulness of the Jones representations\.InGroups of diffeomorphisms,Adv\. Stud\. Pure Math\., Vol\.52,pp\. 369–381\.External Links:ISBN 978\-4\-931469\-48\-8,[Document](https://dx.doi.org/10.2969/aspm/05210369),[Link](https://doi-org.ezproxy1.lib.gla.ac.uk/10.2969/aspm/05210369),[MathReview \(Bruno P\. Zimmermann\)](https://www.ams.org/mathscinet-getitem?mr=2509716)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.SS0.SSS0.Px1.p2.1)\.
- \[LS05\]S\. J\. Lee and W\. T\. Song\(2005\)The kernel ofBurau​\(4\)⊗Zp\{\\rm Burau\}\(4\)\\otimes Z\_\{p\}is all pseudo\-Anosov\.Pacific J\. Math\.219\(2\),pp\. 303–310\.External Links:ISSN 0030\-8730,1945\-5844,[Document](https://dx.doi.org/10.2140/pjm.2005.219.303),[Link](https://doi-org.ezproxy1.lib.gla.ac.uk/10.2140/pjm.2005.219.303),[MathReview \(Pierre A\. Lochak\)](https://www.ams.org/mathscinet-getitem?mr=2175118)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.SS0.SSS0.Px2.p4.6)\.
- \[LP93\]D\. D\. Long and M\. Paton\(1993\)The Burau representation is not faithful forn≥6n\\geq 6\.Topology32\(2\),pp\. 439–447\.External Links:ISSN 0040\-9383,[Document](https://dx.doi.org/10.1016/0040-9383%2893%2990030-Y),[Link](https://doi.org/10.1016/0040-9383(93)90030-Y),[MathReview \(Athanase Papadopoulos\)](https://www.ams.org/mathscinet-getitem?mr=1217079)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p2.5),[§2](https://arxiv.org/html/2607.05283#S2.p11.23),[§2](https://arxiv.org/html/2607.05283#S2.p7.18)\.
- \[LON86\]D\. D\. Long\(1986\)A note on the normal subgroups of mapping class groups\.Math\. Proc\. Cambridge Philos\. Soc\.99\(1\),pp\. 79–87\.External Links:[Document](https://dx.doi.org/10.1017/S0305004100063957),ISSN 0305\-0041,[Link](https://doi-org.ezproxy.lib.gla.ac.uk/10.1017/S0305004100063957),[MathReview \(Bruno P\. Zimmermann\)](https://www.ams.org/mathscinet-getitem?mr=809501)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.SS0.SSS0.Px2.p1.15)\.
- \[MP69\]W\. Magnus and A\. Peluso\(1969\)On a theorem of V\. I\. Arnold\.Comm\. Pure Appl\. Math\.22,pp\. 683–692\.External Links:[Document](https://dx.doi.org/10.1002/cpa.3160220508),ISSN 0010\-3640,[Link](https://doi-org.ezproxy.lib.gla.ac.uk/10.1002/cpa.3160220508),[MathReview \(F\. Gherardelli\)](https://www.ams.org/mathscinet-getitem?mr=264062)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.SS0.SSS0.Px2.p2.5),[§1](https://arxiv.org/html/2607.05283#S1.p2.5),[Theorem 4\.1](https://arxiv.org/html/2607.05283#S4.Thmtheorem1)\.
- \[MAR19\]D\. Margalit\(2019\)Problems, questions, and conjectures about mapping class groups\.InBreadth in contemporary topology,Proc\. Sympos\. Pure Math\., Vol\.102,pp\. 157–186\.External Links:ISBN 978\-1\-4704\-4249\-1,[Document](https://dx.doi.org/10.1090/pspum/102/12),[Link](https://doi-org.ezproxy2.lib.gla.ac.uk/10.1090/pspum/102/12),[MathReview \(Nick Salter\)](https://www.ams.org/mathscinet-getitem?mr=3967367)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.
- \[MOO91\]J\. A\. Moody\(1991\)The Burau representation of the braid groupBnB\_\{n\}is unfaithful for largenn\.Bull\. Amer\. Math\. Soc\. \(N\.S\.\)25\(2\),pp\. 379–384\.External Links:[Document](https://dx.doi.org/10.1090/S0273-0979-1991-16080-5),ISSN 0273\-0979,[Link](https://doi-org.ezproxy.lib.gla.ac.uk/10.1090/S0273-0979-1991-16080-5),[MathReview \(Darren D\. Long\)](https://www.ams.org/mathscinet-getitem?mr=1098347)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p2.5),[§2](https://arxiv.org/html/2607.05283#S2.p11.12),[§2](https://arxiv.org/html/2607.05283#S2.p7.23)\.
- \[MOR91\]S\. Moran\(1991\)Some free groups of matrices and the Burau representation ofB4B\_\{4\}\.Math\. Proc\. Cambridge Philos\. Soc\.110\(2\),pp\. 225–228\.External Links:ISSN 0305\-0041,1469\-8064,[Document](https://dx.doi.org/10.1017/S0305004100070298),[Link](https://doi-org.ezproxy1.lib.gla.ac.uk/10.1017/S0305004100070298),[MathReview \(Darren D\. Long\)](https://www.ams.org/mathscinet-getitem?mr=1113420)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p5.4)\.
- \[WHI00\]Whittlesey, Kim\(2000\)Normal all pseudo\-Anosov subgroups of mapping class groups\.Geom\. Topol\.4,pp\. 293–307\.Cited by:[§1](https://arxiv.org/html/2607.05283#S1.SS0.SSS0.Px2.p4.6)\.
- \[WZ15\]S\. Witzel and M\. C\. B\. Zaremsky\(2015\)A free subgroup in the image of the 4\-strand Burau representation\.J\. Knot Theory Ramifications24\(12\),pp\. 1550065, 16\.External Links:ISSN 0218\-2165,1793\-6527,[Document](https://dx.doi.org/10.1142/S0218216515500650),[Link](https://doi.org/10.1142/S0218216515500650),[MathReview \(Anton Kaul\)](https://www.ams.org/mathscinet-getitem?mr=3415244)Cited by:[§1](https://arxiv.org/html/2607.05283#S1.p3.2)\.