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The reduced Burau representation is faithful for at most three strands

Statement

Assume AC (inherited through the definition of the reduced representation and the agreement theorem with the topological representation). For $1 \leq n \leq 3$ the reduced Burau representation ${\overset{ˉ}{ρ}}_{n} : B_{n} \to {GL}_{n - 1} (Λ_{1})$ over $Λ_{1} = Z [t^{\pm 1}]$ is faithful; that is, $\ker {\overset{ˉ}{ρ}}_{n}$ is trivial and ${\overset{ˉ}{ρ}}_{n}$ is injective. In detail: $B_{1}$ is trivial; $B_{2} ≅ Z$ is generated by $σ_{1}$ (The two-strand braid group is infinite cyclic) and ${\overset{ˉ}{ρ}}_{2} (σ_{1}^{k}) = (- t)^{k}$ , which is $I_{1}$ only for $k = 0$ (Units, powers and the domain property of the Laurent polynomial ring); and if $β \in \ker {\overset{ˉ}{ρ}}_{3}$ , then its specialization at $t = - 1$ lies in $\ker {\overset{ˉ}{ρ}}_{3} (- 1) = ⟨ Δ^{4} ⟩$ (The minus-one specialization of three-strand Burau has kernel generated by Delta to the fourth), say $β = Δ^{4 k}$ , and The reduced Burau representation detects every power of the fourth power of the half twist in B3 forces $t^{6 k} = 1$ , hence $k = 0$ and $β = 1$ . The case $n = 4$ is not claimed. Magnus and Peluso established faithfulness for $n = 3$ by a direct algebraic computation, and the argument for $n = 3$ given here, via the $t = - 1$ specialization, is independent of theirs. The AC hypothesis is exactly the inherited one.

Facts & Assumptions

Given: AC; the ring $Λ_{1} = Z [t^{\pm 1}]$ ; the reduced representations ${\overset{ˉ}{ρ}}_{1}, {\overset{ˉ}{ρ}}_{2}, {\overset{ˉ}{ρ}}_{3}$ in their fixed bases; the half twist $Δ = σ_{1} σ_{2} σ_{1}$ of $B_{3}$ .

[F1]

For $n = 0$ and $n = 1$ there are no Artin generators and $B_{n}$ is the trivial group (The braid group by Artin presentation).

[F2]

$B_{2}$ is infinite cyclic generated by $σ_{1}$ ; in the one-element basis of the reduced module for $n = 2$ the representation is ${\overset{ˉ}{ρ}}_{2} (σ_{1}) = - t$ and ${\overset{ˉ}{ρ}}_{2} (σ_{1}^{k}) = (- t)^{k}$ (The two-strand braid group is infinite cyclic, The topological and matrix Burau representations agree, clause (2); The reduced Burau representation).

[F3]

${\overset{ˉ}{ρ}}_{3} (- 1) : B_{3} \to {GL}_{2} (Z)$ denotes the homomorphism obtained by evaluating the matrices of ${\overset{ˉ}{ρ}}_{3}$ at $t = - 1$ ; its kernel is $\ker {\overset{ˉ}{ρ}}_{3} (- 1) = ⟨ Δ^{4} ⟩ = {Δ^{4 k} : k \in Z}$ (The minus-one specialization of three-strand Burau has kernel generated by Delta to the fourth).

[F4]

${\overset{ˉ}{ρ}}_{3} (Δ^{4 k}) = t^{6 k} I_{2}$ for every $k \in Z$ , and $t^{6 k} I_{2} = I_{2}$ if and only if $k = 0$ (The reduced Burau representation detects every power of the fourth power of the half twist in B3).

[F5]

$t^{m} \neq 1$ in $Λ_{1}$ for every $m \neq 0$ , and in particular $(- t)^{k} = 1$ implies $k = 0$ : if $(- t)^{k} = 1$ then $t^{2 k} = ((- t)^{k})^{2} = 1$ , so $2 k = 0$ (Units, powers and the domain property of the Laurent polynomial ring, clause (b)).

Proof

technique · direct

1.1F1

The case $n = 1$ . By [F1] the group $B_{1}$ is trivial, so its only element is the identity and $\ker {\overset{ˉ}{ρ}}_{1} = {1}$ ; hence ${\overset{ˉ}{ρ}}_{1}$ is injective and faithful.

1.2F2F5algebra

The case $n = 2$ . By [F2] every element of $B_{2}$ is $σ_{1}^{k}$ for a unique $k \in Z$ , and ${\overset{ˉ}{ρ}}_{2} (σ_{1}^{k}) = (- t)^{k}$ . If $σ_{1}^{k} \in \ker {\overset{ˉ}{ρ}}_{2}$ then $(- t)^{k} = 1$ , so $t^{2 k} = 1$ and hence $2 k = 0$ by [F5], giving $k = 0$ and $σ_{1}^{k} = 1$ . Thus $\ker {\overset{ˉ}{ρ}}_{2}$ is trivial and ${\overset{ˉ}{ρ}}_{2}$ is faithful.

1.3F3F4

The case $n = 3$ . Let $β \in \ker {\overset{ˉ}{ρ}}_{3}$ , so ${\overset{ˉ}{ρ}}_{3} (β) = I_{2}$ . Applying the evaluation homomorphism of [F3] entrywise gives ${\overset{ˉ}{ρ}}_{3} (- 1) (β) = I_{2}$ , so $β \in \ker {\overset{ˉ}{ρ}}_{3} (- 1) = ⟨ Δ^{4} ⟩$ , say $β = Δ^{4 k}$ with $k \in Z$ . Then [F4] gives $I_{2} = {\overset{ˉ}{ρ}}_{3} (β) = {\overset{ˉ}{ρ}}_{3} (Δ^{4 k}) = t^{6 k} I_{2}$ , so $t^{6 k} = 1$ and hence $k = 0$ ; therefore $β = Δ^{0} = 1$ . Since $β$ was an arbitrary element of the kernel, $\ker {\overset{ˉ}{ρ}}_{3}$ is trivial and ${\overset{ˉ}{ρ}}_{3}$ is faithful.

2.1step 1.1step 1.2step 1.3∎

Conclusion. Steps 1.1, 1.2 and 1.3 cover $n = 1, 2, 3$ respectively, so the reduced Burau representation is faithful for $1 \leq n \leq 3$ . AC is inherited through the cited representation items as declared; the group-theoretic and specialization computations are choice free.

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Sources

Vasudha Bharathram, Joan S. Birman and Tara E. Brendle, The Burau representation is faithful for n = 4, arXiv:2607.05283v1 (6 July 2026), Introduction and section 4 (Theorem 4.1, the Magnus-Peluso theorem) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey (background on Burau matrices, the cyclic cover and absolute homology) (standard reference, not scraped)
Stephen J. Bigelow, The Burau representation is not faithful for n = 5, Geometry & Topology 3 (1999) 397-404, Theorems 1.2 and 1.4 (printed pp. 397-399) (standard reference, not scraped)