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The image of the full twist under the Burau representation

Example

Assume AC (inherited through the definition of the reduced representation and the agreement theorem with the topological representation). For $n = 3$ let $Δ = σ_{1} σ_{2} σ_{1}$ be the half twist and $Δ^{2}$ the full twist, which generates the center $Z (B_{3})$ (The center of b n is generated by the full twist for n greater than two). Then the reduced Burau representation over $Λ_{1} = Z [t^{\pm 1}]$ sends the full twist to the scalar matrix ${\overset{ˉ}{ρ}}_{3} (Δ^{2}) = t^{3} I_{2},$ so the image of the center is the infinite cyclic subgroup $⟨ t^{3} I_{2} ⟩$ of ${GL}_{2} (Λ_{1})$ , and ${\overset{ˉ}{ρ}}_{3}$ is injective on the center: ${\overset{ˉ}{ρ}}_{3} (Δ^{2 k}) = I_{2}$ holds if and only if $k = 0$ . In particular, although the specialization $t \mapsto - 1$ sends the scalar $t^{3}$ to $- 1$ (so that the image of $Δ^{2}$ becomes $- I_{2}$ and $Δ^{2}$ itself is not in the kernel of ${\overset{ˉ}{ρ}}_{3} (- 1)$ ), neither $Δ^{2}$ nor any $Δ^{2 k}$ with $k \neq 0$ lies in the kernel of ${\overset{ˉ}{ρ}}_{3}$ over $Λ_{1}$ .

Verification

Given: the ring $Λ_{1} = Z [t^{\pm 1}]$ ; the half twist $Δ = σ_{1} σ_{2} σ_{1}$ of $B_{3}$ and the full twist $Δ^{2}$ ; the reduced representation ${\overset{ˉ}{ρ}}_{3} : B_{3} \to {GL}_{2} (Λ_{1})$ in the basis $(g_{1}, g_{2})$ ; the evaluation homomorphism $ε : Λ_{1} \to Z$ , $t \mapsto - 1$ , applied entrywise.

[A1] In the basis $(g_{1}, g_{2})$ , ${\overset{ˉ}{ρ}}_{3} (σ_{1}) = (\begin{array}{cc} - t & t \\ 0 & 1 \end{array})$ , ${\overset{ˉ}{ρ}}_{3} (σ_{2}) = (\begin{array}{cc} 1 & 0 \\ 1 & - t \end{array})$ and ${\overset{ˉ}{ρ}}_{3} (Δ^{2}) = t^{3} I_{2}$ ; ${\overset{ˉ}{ρ}}_{3}$ is a group homomorphism, so ${\overset{ˉ}{ρ}}_{3} (β^{m}) = {\overset{ˉ}{ρ}}_{3} (β)^{m}$ for every $β \in B_{3}$ and every $m \in Z$ (The topological and matrix Burau representations agree, The reduced Burau representation).

[A2] The center of $B_{3}$ is infinite cyclic and generated by the full twist: $Z (B_{3}) = ⟨ Δ^{2} ⟩ = {Δ^{2 k} : k \in Z}$ , with $Δ = σ_{1} σ_{2} σ_{1}$ the half twist (The center of b n is generated by the full twist for n greater than two, The Garside half twist and simple positive braids).

[A3] $t^{m} \neq 1$ in $Λ_{1}$ for every integer $m \neq 0$ (Units, powers and the domain property of the Laurent polynomial ring, clause (b)); in particular $t^{3 k} \neq 1$ for $k \neq 0$ , and $t^{3} I_{2}$ has infinite order in ${GL}_{2} (Λ_{1})$ .

[A4] The assignment $t \mapsto - 1$ extends uniquely to a unital ring homomorphism $ε : Λ_{1} \to Z$ with $ε (t^{m}) = (- 1)^{m}$ , and composition with $ε$ entrywise sends a homomorphism into ${GL}_{2} (Λ_{1})$ to one into ${GL}_{2} (Z)$ (The Laurent polynomial ring as the principal localisation of Z[t] at t, Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ).

Proof technique: direct.

1.1A1A2algebra

The full twist and its powers. By [A1] and $Δ^{2} = (σ_{1} σ_{2} σ_{1})^{2}$ , the full twist satisfies ${\overset{ˉ}{ρ}}_{3} (Δ^{2}) = t^{3} I_{2}$ ; hence for every $k \in Z$ the homomorphism property gives ${\overset{ˉ}{ρ}}_{3} (Δ^{2 k}) = (t^{3} I_{2})^{k} = t^{3 k} I_{2}$ .

2.1A2A3step 1.1

The image of the center. Since $Z (B_{3}) = ⟨ Δ^{2} ⟩$ by [A2], the image of the center is $⟨ {\overset{ˉ}{ρ}}_{3} (Δ^{2}) ⟩ = ⟨ t^{3} I_{2} ⟩$ . The matrix $t^{3} I_{2}$ has infinite order by [A3], so $⟨ t^{3} I_{2} ⟩$ is infinite cyclic. Moreover ${\overset{ˉ}{ρ}}_{3} (Δ^{2 k}) = t^{3 k} I_{2} = I_{2}$ holds if and only if $t^{3 k} = 1$ , which by [A3] happens if and only if $3 k = 0$ , that is, if and only if $k = 0$ ; hence ${\overset{ˉ}{ρ}}_{3}$ is injective on $Z (B_{3})$ .

3.1A1A4step 2.1∎

The specialization at $t = - 1$ . Apply the homomorphism $ε$ of [A4] entrywise to ${\overset{ˉ}{ρ}}_{3} (Δ^{2}) = t^{3} I_{2}$ : the result is ${\overset{ˉ}{ρ}}_{3} (- 1) (Δ^{2}) = (- 1)^{3} I_{2} = - I_{2} \neq I_{2}$ ; so $Δ^{2} \notin \ker {\overset{ˉ}{ρ}}_{3} (- 1)$ , even though its square $Δ^{4}$ has image $(- I_{2})^{2} = I_{2}$ . Over $Λ_{1}$ the same conclusion is step 2.1: ${\overset{ˉ}{ρ}}_{3} (Δ^{2 k}) \neq I_{2}$ for every $k \neq 0$ , so neither $Δ^{2}$ nor any nonzero power $Δ^{2 k}$ lies in the kernel of ${\overset{ˉ}{ρ}}_{3}$ . AC is inherited through the cited agreement theorem; the scalar and matrix computations are choice free.

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Sources

Joan S. Birman and Tara E. Brendle, Braids: A Survey (background on Burau matrices, the cyclic cover and absolute homology) (standard reference, not scraped)
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 2 (printed pp. 1-5) (standard reference, not scraped)