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The Burau determinant recovers the Alexander polynomial of a closed braid

Statement

Assume AC. Let $n \geq 2$ , let $β \in B_{n}$ with closure $\hat{β}$ , and let ${\overset{ˉ}{ρ}}_{n} : B_{n} ⟶ {GL}_{n - 1} (Λ_{1}), Λ_{1} = Z [t^{\pm 1}],$ be the reduced Burau representation of The reduced Burau representation. Then the one-variable Alexander polynomial of $\hat{β}$ of The Alexander polynomial from the zeroth elementary ideal is given, up to multiplication by a unit $\pm t^{m}$ of $Λ_{1}$ , by $Δ_{\hat{β}} (t) ≐ \frac{(1 - t) \det (I_{n - 1} - {\overset{ˉ}{ρ}}_{n} (β))}{1 - t^{n}},$ and if $\hat{β}$ is a knot this equals $\det (I_{n - 1} - {\overset{ˉ}{ρ}}_{n} (β)) / (1 + t + \dots + t^{n - 1})$ up to units. In particular for $n = 2$ and $β = σ_{1}^{m}$ one has ${\overset{ˉ}{ρ}}_{2} (σ_{1}^{m}) = (- t)^{m}$ and $Δ_{\hat{σ_{1}^{m}}} (t) ≐ \frac{(1 - t) (1 - (- t)^{m})}{1 - t^{2}},$ so for $m = 3$ this is $t^{2} - t + 1$ , the trefoil value. The formula computes the oriented link invariant $Δ_{\hat{β}}$ from any braid representative of the link, with the stated unit ambiguity.

Facts & Assumptions

Given: an integer $n \geq 2$ , a braid $β \in B_{n}$ , its closure $\hat{β}$ , the reduced Burau representation ${\overset{ˉ}{ρ}}_{n}$ and the coloured reduced Burau matrix ${\overline{B}}_{β} (t_{1}, \dots, t_{n})$ with equal-label specialisation $B_{β} (t)$ . AC is inherited from the reduced Burau representation and Alexander-module suppliers.

[F1]

The reduced Burau module $M_{r e d}$ is free with the auxiliary basis $h_{j} = ε_{j} - ε_{n}$ ( $h_{n} = 0$ ), and ${\overset{ˉ}{ρ}}_{n} (β)$ is the action matrix in the fixed basis $b_{j} = t^{j} (h_{j} - h_{j + 1})$ , not in the $h$ basis (The reduced Burau module is free of rank n minus one, The reduced Burau representation).

[F2]

The unreduced Burau matrices of The unreduced Burau matrices are the matrices $B_{i}$ that are the identity outside rows and columns $i, i + 1$ , with block $(\begin{array}{cc} 1 - t & t \\ 1 & 0 \end{array})$ , acting on column vectors in the relative lifted-edge basis $e_{1}, \dots, e_{n}$ , and the topological action of $B_{n}$ on $U = H_{1} (\tilde{X}, p^{- 1} d; Z)$ in that basis is this matrix representation (The topological and matrix Burau representations agree).

[F3]

$M_{r e d}$ is the kernel of the connecting map $\partial_{*} : U \to Λ_{1}$ , and in the relative basis $\partial_{*} (e_{i}) = t^{i - 1} (t - 1)$ , so that the invariant covector is $σ (e_{i}) = t^{i - 1}$ and $M_{r e d} = \ker σ$ ; the exact sequence $0 \to M_{r e d} \to U \to Λ_{1} \to Z \to 0$ is $B_{n}$ -equivariant, so the action on $M_{r e d}$ is the restriction of the action on $U$ . The level- $0$ class $ε_{i}$ of the $i$ -th lifted edge satisfies $ε_{i} = t^{- (i - 1)} e_{i}$ (The unreduced module fits an exact sequence with the reduced module, The unreduced Burau matrices, The reduced Burau module is free of rank n minus one).

[F4]

The coloured reduced Burau matrix at equal labels $B_{β} (t) = {\overline{B}}_{β} (t, \dots, t)$ is the product of the matrices ${\overline{C}}_{i} (t)^{\pm 1}$ of The coloured reduced Burau matrix along an Artin word for $β$ , where ${\overline{C}}_{i} (t)$ has $i$ -th row entries $t$ at $(i, i - 1)$ , $- t$ at $(i, i)$ and $1$ at $(i, i + 1)$ , truncated at the boundary columns.

[F5]

The Burau determinant formula of The Burau determinant formula for a closed braid and its axis(2): in the one-variable specialisation the one-variable Alexander polynomial satisfies $Δ_{\hat{β}} (t) ≐ (1 - t) \det (I - B_{β} (t)) / (1 - t^{n})$ , equivalently $D_{\hat{β}} (t) ≐ \det (I - B_{β} (t)) / (1 - t^{n})$ for a knot; the identifications of the strand variables are those of the closed braid.

[F6]

$Δ_{\hat{β}}$ is an invariant of the oriented link type of $\hat{β}$ , well defined up to multiplication by a unit $\pm t^{m}$ (The Alexander polynomial is an oriented link invariant, The Alexander polynomial from the zeroth elementary ideal).

Proof

1.1F1F2F3algebra

The generators in the reduced basis. Put $h_{j} : = ε_{j} - ε_{n} = t^{- (j - 1)} e_{j} - t^{- (n - 1)} e_{n}$ for $1 \leq j \leq n - 1$ , the basis of [F1, F3]. For $1 \leq i \leq n - 2$ the matrix of $σ_{i}$ in this basis is the identity except for the block $(\begin{array}{cc} 1 - t & 1 \\ t & 0 \end{array})$ in rows and columns $i, i + 1$ ; for $i = n - 1$ it is the identity except for the last row $(- 1, - 1, \dots, - 1, - t)$ . Both assertions are the finite computation $B_{i} h_{j} = \sum_{k} (R_{i})_{k j} h_{k}$ using $B_{i} h_{j} \in \ker σ$ and the expression of the result in the basis $h$ , carried out on the two or three vectors moved by $B_{i}$ .

2.1F1F2F3F4step 1.1algebra

Conjugation with the equal-label matrices. Put $P = I_{n - 1} - S$ , where $S$ is the subdiagonal shift, and $D = diag (t, t^{2}, \dots, t^{n - 1})$ . The columns of $Q = P D$ are the coordinates of the fixed basis $b_{j} = t^{j} (h_{j} - h_{j + 1})$ in the auxiliary $h$ basis, so ${\overset{ˉ}{ρ}}_{n} (σ_{i}) = Q^{- 1} R_{i} Q$ . The matrices of step 1.1 satisfy $R_{i} P = P {\overline{C}}_{i} (t)$ : for $i < n - 1$ this is multiplication of the displayed two-row block; for $i = n - 1$ the last row of $R_{i} P$ is zero before column $n - 2$ , then $t - 1, - t$ , as in $P {\overline{C}}_{i} (t)$ . At $n = 2$ the identity is the scalar $- t = - t$ . Hence ${\overset{ˉ}{ρ}}_{n} (σ_{i}) = D^{- 1} {\overline{C}}_{i} (t) D$ . Multiplying these identities, including inverses, along the word gives ${\overset{ˉ}{ρ}}_{n} (β) = D^{- 1} {\overline{B}}_{β} (t) D$ , and therefore $\det (I_{n - 1} - {\overline{B}}_{β} (t)) = \det (I_{n - 1} - {\overset{ˉ}{ρ}}_{n} (β))$ . This also agrees with the frozen generator formulas of The topological and matrix Burau representations agree.

3.1F5F6step 2.1algebra

The Alexander formula. Substituting the determinant identity of step 2.1 into the formula of [F5] gives $Δ_{\hat{β}} (t) ≐ (1 - t) \det (I_{n - 1} - {\overset{ˉ}{ρ}}_{n} (β)) / (1 - t^{n})$ , which is the displayed formula. For a knot, dividing by $1 - t$ and using $1 - t^{n} = (1 - t) (1 + t + \dots + t^{n - 1})$ gives $D_{\hat{β}} (t) ≐ \det (I_{n - 1} - {\overset{ˉ}{ρ}}_{n} (β)) / (1 - t^{n})$ , equivalently $Δ_{\hat{β}} (t) ≐ \det (I_{n - 1} - {\overset{ˉ}{ρ}}_{n} (β)) / (1 + t + \dots + t^{n - 1})$ , up to units. The right-hand side is computed from any braid representative of the link, while the left-hand side is the oriented link invariant of [F6]; this also shows that the right-hand side does not depend on the representative, up to the stated unit.

4.1F5step 1.1step 3.1algebra∎

The two-strand case. For $n = 2$ the module $M_{r e d} = \ker σ$ is one-dimensional with basis $h_{1} = e_{1} - t^{- 1} e_{2}$ and, by step 1.1, ${\overset{ˉ}{ρ}}_{2} (σ_{1}) = - t$ ; hence ${\overset{ˉ}{ρ}}_{2} (σ_{1}^{m}) = (- t)^{m}$ and the formula becomes $Δ_{\hat{σ_{1}^{m}}} ≐ \frac{(1 - t) (1 - (- t)^{m})}{1 - t^{2}}$ . For $m = 1$ this is $≐ 1$ (the unknot); for $m = 3$ it is $\frac{(1 - t) (1 + t^{3})}{1 - t^{2}} = \frac{1 + t^{3}}{1 + t} = t^{2} - t + 1$ , the trefoil value; and for $m = 2$ the closure $\hat{σ_{1}^{2}}$ has two components and the same display gives $Δ ≐ (1 - t) (1 - t^{2}) / (1 - t^{2}) = 1 - t$ , the one-variable Alexander polynomial of the Hopf link in the convention of [F5]. These computations prove the displayed specialisations of the statement.

Remarks

The deps of this proposition include four items of the sibling pair the-burau-representations (The unreduced Burau matrices, The unreduced module fits an exact sequence with the reduced module, The reduced Burau module is free of rank n minus one, The topological and matrix Burau representations agree). They supply the matrix realization of the abstract reduced representation used in steps 1.1–2.1; their current statements supply exactly the relative basis, invariant covector and topological action used here.
The design listed Markov's theorem among the prerequisites of this item; it is not needed, because the identity is proved for each braid representative directly from the determinant theorem, and the invariance of the left-hand side is The Alexander polynomial is an oriented link invariant.

Depends on

Used by

The Burau determinant for a two-strand torus link Example

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64 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.2 equation (15) (printed p. 47): the reduced Burau determinant formula for the Alexander polynomial (standard reference, not scraped)
H. R. Morton, The multivariable Alexander polynomial for a closed braid, arXiv:math/9803138, Remark (1) and Remark (2) (printed pp. 3-4) (standard reference, not scraped)