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

Joan S. Birman; Tara E. Brendle; Vasudha Bharathram

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

Vasudha Bharathram, Joan S. Birman, Tara E. Brendle

Geometric Topology · Submitted 6 Jul 2026

Pith's verdict: Resolves the last open case of Burau faithfulness (n=4); the proof strategy is sound but Proposition 6.4's geometric case analysis needs careful verification.

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CitationJoan S. Birman, Tara E. Brendle, Vasudha Bharathram. "The Burau representation of the braid group is faithful for n = 4." arXiv:2607.05283 (2026). https://pith.science/paper/AR6VZSKQ

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The headline: this paper claims faithfulness of the Burau representation for B_4, the last open case in a problem dating to 1935. The result also implies faithfulness of the Jones representation for B_4. This is a significant result for geometric topology and knot theory, and the proof strategy is coherent and well-motivated throughout. The paper is by experienced hands — Birman and Brendle are deeply established in this area — and the approach builds naturally on the existing toolkit (Moody polynomials, Long's theorem, Bigelow's winding numbers) rather than inventing machinery from scratch. The new ingredients — the parity condition (Section 4), proper products of push-maps (Section 5), and the B_5 correction trick (Section 6.3) — are clean ideas that fit the problem well. The new proof of the n=3 case (Theorem 4.1) via the parity condition is a nice bonus that makes the paper self-contained and illustrates the method before the harder case. The logical structure is clear: reduce to Brun_4 via Long's theorem, conjugate to get a proper product, embed in B_5 to correct parity, apply Moody's obstruction. No circularity; the external results are used correctly. The soft spots are concentrated in two places. Proposition 5.1 involves a dense case analysis of bigon-forming configurations that is geometrically intuitive but not fully rigorous in every sub-case — the reader has to fill in connectivity arguments about complements that are stated but not verified. The more serious concern is Proposition 6.4, which constructs the correcting push-map Gamma. The stress-test flag here is legitimate: when the proof modifies gamma_2 to gamma'_2 in the second stage, it checks that the modification preserves the intersection inequality and the first-intersection ordering, but it does not explicitly verify that Gamma' still satisfies the parity condition. The new segments gamma' and gamma'' can create new disks in the disk sequence of (beta_3*)·Gamma' that are not analyzed. If one of those new disks violates parity, the chain from Corollary 6.5 to Theorem 6.6 breaks, because Lemma 4.2 (no cancellations) requires the parity condition. This is the load-bearing gap. It may well be fixable — the construction is geometrically plausible and the authors clearly understand the constraints — but as written, the verification is incomplete. The reader's CONDITIONAL verdict and MODERATE confidence are about right. This is not a paper with a flawed strategy; it is a paper with a correct strategy and one under-verified step in its most technical proposition. Who gets value: anyone working on braid group representations, mapping class groups, or the Burau/Jones connection. The methods may also be useful for the open questions about characterizing the kernel and image of Burau, as the authors note. This deserves a serious referee — ideally someone comfortable with the geometric topology of arcs and disks in punctured disks who can either verify the case analysis in Proposition 6.4 or identify where it needs shoring up. The n=3 proof and the overall framework are already solid contributions even if the n=4 case needs revision.

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Major objections (4)

Proposition 6.4, second stage of the proof (construction of Γ' = γ₁ ∪ γ'₂ where γ'₂ = γ' ∪ γ'' ∪ γ₂): The proof verifies two properties of γ'₂ — (1) its first intersection with β occurs after its final intersection with α, and (2) ι(α, (β₃*)·Γ') > ι(α, (β₃*)·Γ) — and then concludes 'we have established the result in this case.' However, the full statement of Proposition 6.4 requires that Γ' also satisfies the parity condition. The modification inserts new segments γ' and γ'' (where γ' terminates at a point r ∈ α) into the push-map loop. These new segments can create new disks in the disk sequence of (β₃*)·Γ' that were not present in the disk sequence of (β₃*)·Γ. The proof does not analyze what types of disks these new segments create, nor verify that they satisfy the parity condition. This matters because Corollary 6.5 uses the parity condition for Γ to conclude that M_Γ has no cancellat
Proposition 6.4, first stage of the proof (construction of Γ = γ₁ ∪ γ₂): The verification that Γ itself satisfies the parity condition relies on the claim that 'each α-γ₁ disk is equivalent to one of the five disks shown in Figure 6.5; otherwise we could have made a different choice of γ₁.' This is a crucial step, as the parity condition for Γ is load-bearing for Corollary 6.5. The argument that a suitable γ₁ can always be chosen so that all resulting α-γ₁ disks fall into the five types of Figure 6.5 is stated without rigorous verification. The claim depends on connectivity properties of the complement D₄ ∖ (β ∪ γ₁) and on the interaction of γ₁ with the specific loop Γ₁ from Figure 6.3. A more detailed justification — or at minimum a clear statement of why the 'otherwise we could have made a different choice' argument always succeeds — is needed.
Proposition 6.4, Case 2 of the second stage: The proof states that D₄ ∖ (β ∪ γ₁ ∪ γ') is 'homeomorphic to a disk with at least one puncture' and hence a nontrivial loop based at r exists. However, the existence of a nontrivial loop does not, by itself, guarantee that the resulting γ'' can be chosen to satisfy the disjointness conditions (disjoint from β, γ'' ∩ (γ₁ ∪ γ') = {q, r}, not isotopic to γ') while also ensuring that the modified Γ' = γ₁ ∪ γ'₂ still satisfies the parity condition. The interaction between the new loop and the disk sequence of (β₃*)·Γ' is not analyzed. This gap compounds the concern in the first major comment above.
Theorem 6.6: The proof argues that any nontrivial Φ ∈ Brun₄ can be conjugated so that Φ·Γ₁ is a proper product, using the free basis of K₄ shown in Figure 6.9. The argument is that Γ₁ begins with y₂⁻¹, so if Φ ends with y₂, we replace Φ with a conjugate that does not. However, the proof does not explicitly verify that such a conjugate can always be found within Brun₄ (rather than just within K₄ or B₄). Since the reduction to Brun₄ via Long's theorem (Proposition 1.2) requires faithfulness on Brun₄ specifically, the conjugation must preserve membership in Brun₄. A brief argument clarifying this point would strengthen the proof.

Minor objections (7)

Abstract: 'faithful forn= 4' — missing space before 'n'.
Section 1, first paragraph: 'faithfulness of the Burau representation forn= 3' — missing space.
Section 2, first paragraph: 'the diskD n' — missing space throughout; this spacing issue recurs frequently (e.g., 'D n', 'B n', 'p ∗').
Theorem 2.3 proof: 'Since Γ−1 ·Φ·Γ acts trivially on this homology group' — it would help to explicitly note that this follows from the assumption that Φ ∈ ker(ρ_n), so Γ⁻¹·Φ·Γ ∈ ker(ρ_n) as well.
Figure 6.5: The five disk types δ₁,...,δ₅ are referenced in the proof of Proposition 6.4 but the figure labels are small and hard to distinguish. Consider enlarging or adding a legend.
Section 7: The example uses an arc β that 'does not arise as (β₃*)Φ for any choice of Φ ∈ K₄.' While the authors explain this is for illustration, it would be more convincing to include (or at least reference) a computation for a genuine proper product arising from Brun₄.
The remark following Proposition 6.4 about generalization to arbitrary n is interesting but somewhat speculative. Consider clarifying whether this is a formal conjecture or an informal observation.

Simulated authors' rebuttal

read the rebuttal and 4 point-by-point responses

The referee identifies a genuine gap in the verification of the parity condition for the modified push-map Γ' in the second stage of Proposition 6.4, along with related concerns about the rigor of the first stage and the conjugation argument in Theorem 6.6. We agree that Comments 1–3 require substantive revision and that Comment 4 requires a clarifying addition. We believe all four can be addressed, but the parity verification for Γ' (Comments 1 and 3) will require the most significant work.

Referee: Proposition 6.4, second stage: the proof verifies two properties of γ'₂ but does not verify that Γ' satisfies the parity condition. New segments γ' and γ'' may create new disks whose types are not analyzed.

Authors: The referee is correct. In the second stage of the proof of Proposition 6.4, we construct Γ' = γ₁ ∪ γ'₂ (where γ'₂ = γ' ∪ γ'' ∪ γ₂) and verify two properties — the ordering condition on intersections with β and α, and the inequality ι(α, (β₃*)·Γ') > ι(α, (β₃*)·Γ) — but we do not explicitly verify that Γ' satisfies the parity condition. This is a genuine gap, because the new segments γ' and γ'' (which together form a loop based at q, traveling through a point r ∈ α) can create new α-Γ' disks in the disk sequence of (β₃*)·Γ' that were not present in the disk sequence of (β₃*)·Γ. We will revise the proof to include a detailed analysis of these new disks. The key observation we will make explicit is that γ' and γ'' are chosen to be disjoint from β = (β₃*)·Φ, and the loop γ' ∪ γ'' lies in D₄ ∖ (β ∪ γ₁), which is a disk with at least one puncture. The new α-Γ' disks created by γ' and γ'' are therefore constrained: they arise from the interaction of the loop γ' ∪ γ'' with α and β₃* (not β), and we will show that each such disk is equivalent to one of the five types in Figure 6.5 or is a 1-disk containing only p₅, by the same type of argument used in the first stage. We acknowledge that this analysis was implicit in our original argument but was not written out, and we will add it in the revision. revision: yes
Referee: Proposition 6.4, first stage: the claim that each α-γ₁ disk is equivalent to one of the five disks in Figure 6.5, justified by 'otherwise we could have made a different choice of γ₁,' is stated without rigorous verification.

Authors: The referee is right that this argument is insufficiently justified as written. The claim depends on the fact that γ₁ is a simple arc from p₅ to q that is disjoint from β and intersects α minimally, and that Φ = Φ'·Γ₁ is a proper product. The 'otherwise we could have made a different choice' argument is meant to invoke the proper product condition: since Φ·Γ₁ is a proper product, there are no bigon-forming polygons between α, β, and Γ₁, and this constrains the possible α-γ₁ disks. However, we agree that the connection between the absence of bigon-forming polygons and the restriction to the five disk types of Figure 6.5 is not spelled out. We will revise this portion of the proof to give a more detailed argument. Specifically, we will show that any α-γ₁ disk that is not equivalent to one of the five types in Figure 6.5 would necessarily give rise to a bigon-forming polygon (a generalized bigon, β-to-α trigon, or generalized rectangle) between α, β, and Γ₁, contradicting the proper product hypothesis. This is a finite case analysis depending on the position of the disk relative to Γ₁ and the punctures, and we will include it explicitly. revision: yes
Referee: Proposition 6.4, Case 2 of second stage: existence of a nontrivial loop does not guarantee that γ'' can be chosen to satisfy disjointness conditions while also ensuring Γ' satisfies the parity condition.

Authors: This concern is well-taken and is closely related to the gap identified in the first major comment. In Case 2, we argue that D₄ ∖ (β ∪ γ₁ ∪ γ') is homeomorphic to a disk with at least one puncture, and hence a nontrivial loop based at r exists. The referee correctly points out that the existence of such a loop does not, by itself, guarantee that the resulting γ'' satisfies the disjointness conditions (disjoint from β, γ'' ∩ (γ₁ ∪ γ') = {q, r}, not isotopic to γ') while also preserving the parity condition for Γ'. We will address this in two steps. First, the disjointness conditions are satisfiable because D₄ ∖ (β ∪ γ₁ ∪ γ') has at least one puncture, so there exist nontrivial loops based at r that avoid β and γ₁ ∪ γ' except at their endpoints; this is a standard fact about fundamental groups of punctured disks. Second, and more substantively, we will verify the parity condition for Γ' by analyzing the new disks created by γ'' (and γ'), as discussed in our response to the first major comment. The new disks created by the loop γ' ∪ γ'' are constrained by the fact that this loop lies in D₄ ∖ (β ∪ γ₁), and we will show that each resulting α-Γ' disk is equivalent to one of the five types in Figure 6.5 or is a 1-disk containing only p₅. We acknowledge that this analysis was missing from the original manuscript and will be added. revision: yes
Referee: Theorem 6.6: the conjugation to ensure Φ does not end with y₂ is not explicitly verified to preserve membership in Brun₄.

Authors: This point can be resolved with a brief addition. As noted in the introduction, Brun₄ is a normal subgroup of B₄. Therefore, for any Φ ∈ Brun₄ and any y ∈ B₄, the conjugate yΦy⁻¹ also lies in Brun₄. In the proof of Theorem 6.6, the conjugation is performed by an element of K₄ ⊂ B₄ (specifically, by an element that cancels the terminal y₂ in the free reduction of Φ·Γ₁), so the conjugate remains in Brun₄ by normality. We will add an explicit sentence to the proof of Theorem 6.6 stating that Brun₄ is normal in B₄ and hence the conjugation preserves membership in Brun₄. revision: yes

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The paper

The paper claims

Burau representation proven faithful for 4-strand braids

The reading

The Burau representation assigns a matrix to each braid, and the question of whether this assignment is faithful — meaning different braids always get different matrices — has been settled for all braid sizes except 4. This paper proves that the Burau representation of the 4-strand braid group B_4 is faithful, closing the last open case. The proof reduces the problem to the Brunnian subgroup Brun_4 (braids that become trivial when any one strand is removed), using a theorem of Long that faithfulness on any nontrivial normal subgroup implies faithfulness on the whole group. The authors then show that any nontrivial Brunnian braid can be conjugated into a form that factors as a proper product of push-maps. By embedding the 4-strand braid into a 5-strand braid and constructing a corrective push-map that enforces a parity condition on disk sequences, they apply Moody's criterion to conclude that no nontrivial Brunnian braid lies in the kernel. An immediate corollary is that the Jones representation of B_4 is also faithful, since it contains the Burau representation as a summand.

What carries the argument

The Moody polynomial M(α,β) is an invariant of a pair of arcs in the punctured disk, built from algebraic intersection numbers in the infinite cyclic cover. Moody's theorem states that if M_{Φ·Γ} ≠ M_Γ for some braid Γ, then Φ is not in the kernel of the Burau representation. The disk sequence of a braid records, for each consecutive pair of crossings between the arcs α and β, the number of punctures enclosed and the winding number. The parity condition links the sign-change property of each disk to the parity of its puncture count, ensuring no two terms in the Moody polynomial cancel. Long's theorem reduces faithfulness on B_n to faithfulness on any nontrivial noncentral normal subgroup, in

Extended reading notes

Core claim

The central mechanism is a parity condition on disk sequences: if every disk in a braid's disk sequence is sign-changing exactly when it contains an odd number of punctures, then the braid's Moody polynomial admits no cancellations and hence the braid cannot lie in the kernel of the Burau representation. For 3-strand braids this condition holds automatically. For 4-strand braids it does not, but the authors show that any nontrivial Brunnian 4-braid can be conjugated to admit a proper product factorization, embedded into B_5, and corrected by a push-map so that both the corrected braid and the correcting map satisfy the parity condition. The push-map replaces 4-puncture disks with 5-puncture,

Load-bearing premise

The proof of Proposition 6.4 constructs a push-map Γ in B_5 that simultaneously replaces every 4-disk in the disk sequence with a 5-disk, preserves all other disk types, satisfies the parity condition itself, and ensures unequal intersection numbers. The construction relies on choosing arcs with specific disjointness and intersection properties, and the claim that such arcs always exist depends on connectivity arguments of complements that are stated but not fully verified in

Original abstract

In this paper we use ideas introduced earlier by Moody, Long, Long-Paton, and Bigelow to prove the theorem of the title, that the Burau representation of the classical braid group is faithful for n = 4. An immediate corollary is that the Jones representation of the braid group is also faithful for n = 4.

Figures

**Figure 1.** 1: A Brunnian 4-braid β. 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. 2

**Figure 2.** 2: The blue arc α joins p1 to p2. The red arc β 3 ∗ joins the basepoint on ∂D to the point p3. representation is equivalent to the existence of an arc β joining the basepoint p∗ to the marked point p3 in Dn such that ι(α, β) > 0 and M(α, β) = 0 [Moo91]. With this in mind, we define the Moody polynomial of the braid Φ ∈ Bn as follows: MΦ = M(α,(β 3 ∗ )Φ). We remark that by taking the boundary of a regular neighborhood of the arc (β 3 ∗ )Φ, we can view as a loop based at p∗ traveling around p3, and that MΦ is well defined on the homology class of a lift of [(β 3 ∗ )Φ] in H1(Dfn,

**Figure 3.** 1: The arcs α and β in (a) give rise to a singleton winding number sequence given by W1 = 1, with corresponding Moody polynomial 1 − t. The winding number sequence for the arcs in (b) is given by 1, 1, −1, −2, 1; its Moody polynomial is 2 − 2t + t 2 − t −1 . Lemma 3.1 (Bigelow [Big99]). Let qi and qi+1 be two points in α ∩ β that are consecutive with respect to the oriented arc β, and let ki and ki+1 be the exponents of the monomials corresponding to qi and qi+1. If Wi denotes the i th term in the winding number sequence for β, then we have: Wi = ki+1 − ki . (1) We record here the fo

**Figure 4.** 1: Possible disks in the disk sequence of an element of B

**Figure 5.** 6: On the left two subarcs of Γ intersect β with the same sign; on the right two subarcs of Γ intersect β with different signs. rectangle between Γ, α, and β, which contradicts our assumption that Φ · Γ is a proper product. If they are different signs, this implies the existence of a generalized bigon between Γ and α, which again contradicts our assumption that Φ · Γ is a proper product. Therefore at least one vertex of any bigon between α and (β)Γ must be an original point of intersection. Now, the support of the push-map Γ is a regular neighborhood of Γ in Dn. Hence we can…

**Figure 6.** 1: A sign-preserving disk 6.2 The point-pushing subgroup K4 Our focus in the remainder of this section will be the point-pushing subgroup K4. Our next step will be to analyze the types of disks that can appear in the disk sequence of certain braids in K4. Specifically, we will be constructing proper products of certain elements in K4 to which we can apply Lemma 4.2. We will then analyze the corresponding disk sequences, which will ultimately provide a mechanism for applying Theorem 2.3. Lemma 6.2. Let Φ ∈ K4, let Γ ∈ K4 be a push-map along a simple loop, and suppose that Φ = Ψ · Γ is…

**Figure 7.** 1: An example of an arc β whose Moody polynomial admits a cancellation. Note that any braid Φ inducing the arc β does 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 of α with β as we travel along β from the basepoint to p3. We compute the Moody polynomial for Φ as follows, noting that the linear terms appear with opposite signs, yielding a cancellation. MΦ = t 0 + t 2 + t 4 − t 3 − t + t −3 + t −1 + t = 1 + t 2 + t 4 − t 3 + t −3 + t −1 . Following the…

Figures reproduced from arXiv:2607.05283 by the authors.

References

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Very recently, Bharathram, Birman, and Brendle proved that it is faithful for $m=4$ ( [ref] ).] $f(b)$ of an $m$-braid $b$ is an $m ×m$ matrix-valued invariant with Laurent polynomial entries that can be computed from the Artin generators ( [ref] ; see also [ref] ).

The crossing matrix and the Burau matrix of pure braids · Introduction · 0.92

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