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The reduced Burau representation

Definition

Assume AC (inherited from the lift of braid mapping classes). Let $M_{r e d}$ be the reduced Burau module over $Λ_{1} = Z [t^{\pm 1}]$ , free of rank $n - 1$ . Write $h_{i} = [ϵ_{i} - ϵ_{n}]$ for the transported basis of The reduced Burau module is free of rank n minus one, and put $h_{n} = 0$ . For the matrix representation fix the adjacent weighted basis $b_{i} : = t^{i} (h_{i} - h_{i + 1}) = [t^{i} (ϵ_{i} - ϵ_{i + 1})], 1 \leq i \leq n - 1.$ It is a basis because $h_{i} = \sum_{k = i}^{n - 1} t^{- k} b_{k}$ ; these formulas are inverse changes of coordinates. For $n = 1$ both bases are empty. Let $\tilde{h}$ be the basepoint-normalised lift of a representative homeomorphism of $β \in B_{n}$ constructed in Braid mapping classes lift equivariantly to the Burau cover. The reduced Burau representation is ${\overset{ˉ}{ρ}}_{n} : B_{n} ⟶ {GL}_{n - 1} (Λ_{1}), β ⟼ the matrix of (\tilde{h})_{*} on M_{r e d} in the fixed basis .$

It is well defined because a different representative is isotopic and its lift acts by the same $Λ_{1}$ -module automorphism, and because the matrix is taken in the basis $(b_{1}, \dots, b_{n - 1})$ ; it is a homomorphism because the induced homology action is a homomorphism; and it is $Λ_{1}$ -linear because the lift commutes with the deck group. Conventions: matrices act on column vectors, with the basis $(b_{i})$ in increasing index order. The original basis $(h_{i})$ gives conjugate matrices.

Facts & Assumptions

Given: AC; the identification $B_{n} ≅ Mod (D^{2}, Q_{n}; \partial D^{2})$ of Braid group as boundary-fixed punctured-disk mapping classes, The Artin presentation is complete for geometric braids; the reduced Burau module $M_{r e d}$ with its fixed $Λ_{1}$ -basis; and a braid $β \in B_{n}$ .

[F1]

For every braid class there is a homeomorphism representative; the basepoint-normalised lifts of isotopic representatives are isotopic as maps of pairs and therefore induce equal homology maps; these induced maps form a homomorphism from $B_{n}$ , and each $\tilde{h}$ acts on $M_{r e d}$ by a $Λ_{1}$ -module automorphism (Braid mapping classes lift equivariantly to the Burau cover, Braid group as boundary-fixed punctured-disk mapping classes, The Artin presentation is complete for geometric braids).

[F2]

$M_{r e d}$ is free of rank $n - 1$ over $Λ_{1}$ with basis $(h_{i})$ ; the inverse coordinate formulas in the definition give the fixed basis $(b_{i})$ . The matrix of a $Λ_{1}$ -module endomorphism in a fixed basis is invertible exactly when the endomorphism is an automorphism; matrices compose under the library convention that the leftmost factor is applied last (The reduced Burau module is free of rank n minus one, Invertible square matrices and similarity over a commutative ring, The Laurent polynomial ring as the principal localisation of Z[t] at t).

Proof

technique · direct

1.1F1F2

Well-definedness. Choose a representative homeomorphism $h$ of the mapping class of $β$ ; any two such representatives are isotopic through boundary-fixed homeomorphisms preserving $Q_{n}$ setwise, so by [F1] their basepoint-normalised lifts have the same action on $M_{r e d}$ . Hence the $Λ_{1}$ -module automorphism $(\tilde{h})_{*}$ depends only on $β$ ; its matrix in the fixed basis therefore depends only on $β$ .

1.2F1F2

Homomorphism property. If $h_{1}, h_{2}$ represent $β_{1}, β_{2}$ , then $\tilde{h_{1} h_{2}} = {\tilde{h}}_{1} \circ {\tilde{h}}_{2}$ by [F1], so $(\tilde{h_{1} h_{2}})_{*} = ({\tilde{h}}_{1})_{*} \circ ({\tilde{h}}_{2})_{*}$ and, in the fixed basis, the matrix of the composite is the product of the matrices in the library composition order of [F2]. Thus ${\overset{ˉ}{ρ}}_{n} (β_{1} β_{2}) = {\overset{ˉ}{ρ}}_{n} (β_{1}) {\overset{ˉ}{ρ}}_{n} (β_{2})$ .

2.1F1F2step 1.1step 1.2∎

$Λ_{1}$ -linearity and target. By [F1] each $(\tilde{h})_{*}$ is a $Λ_{1}$ -module automorphism of the free module $M_{r e d}$ of rank $n - 1$ ; its matrix in the fixed basis is therefore an invertible matrix over $Λ_{1}$ , i.e. an element of ${GL}_{n - 1} (Λ_{1})$ , with inverse the matrix of $({\tilde{h}}^{- 1})_{*}$ . This proves that ${\overset{ˉ}{ρ}}_{n}$ is a well-defined group homomorphism into ${GL}_{n - 1} (Λ_{1})$ . AC enters only through the mapping-class identification and the lift of [F1].

Depends on

Used by

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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), section 2 (printed pp. 1-5) (standard reference, not scraped)