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The Burau determinant for a two-strand torus link

Example

Assume the Axiom of Choice (The Axiom of Choice), as required by the Alexander module and determinant formula. For $β = σ_{1}^{m} \in B_{2}$ the reduced Burau representation is one-dimensional, ${\overset{ˉ}{ρ}}_{2} (σ_{1}^{m}) = (- t)^{m}$ , and the formula of The Burau determinant recovers the Alexander polynomial of a closed braid gives $Δ_{\hat{σ_{1}^{m}}} (t) ≐ \frac{(1 - t) (1 - (- t)^{m})}{1 - t^{2}} .$ For $m = 1$ this is the unknot with $Δ ≐ 1$ ; for $m = 3$ it is the trefoil with $Δ ≐ t^{2} - t + 1$ ; for $m = - 3$ it is the mirror trefoil with $Δ ≐ t^{- 2} - t^{- 1} + 1$ ; and for $m = 2$ it is the Hopf link with $Δ ≐ 1 - t$ . For even $m$ the closure of $σ_{1}^{m}$ has two components (the $(2, m)$ -torus link) and the same display gives the one-variable Alexander polynomial of that link, $Δ ≐ (1 - t) (1 - t^{m}) / (1 - t^{2})$ , which for $m = 2$ is $1 - t$ ; the knot formula is the special case of odd $m$ .

Verification

Given: AC and the braid $β = σ_{1}^{m} \in B_{2}$ , the reduced Burau representation ${\overset{ˉ}{ρ}}_{2} : B_{2} \to {GL}_{1} (Λ_{1})$ , $Λ_{1} = Z [t^{\pm 1}]$ , and the closure $\hat{σ_{1}^{m}}$ . AC is used through the Alexander module and determinant formula; the finite scalar calculations require no further choice.

[A1] $Δ_{\hat{β}} (t) ≐ \frac{(1 - t) \det (I_{n - 1} - {\overset{ˉ}{ρ}}_{n} (β))}{1 - t^{n}}$ for $β \in B_{n}$ , with the unit ambiguity $\pm t^{m}$ (The Burau determinant recovers the Alexander polynomial of a closed braid, The Alexander polynomial from the zeroth elementary ideal).

[A2] For $n = 2$ the reduced module is free of rank one with $h_{1} = [ϵ_{1} - ϵ_{2}]$ and fixed basis $b_{1} = t h_{1}$ (The reduced Burau representation). In that fixed basis ${\overset{ˉ}{ρ}}_{2} (σ_{1}^{m}) = (- t)^{m}$ , as supplied by The Burau determinant recovers the Alexander polynomial of a closed braid.

[A3] The closure of $β \in B_{n}$ has as many components as the permutation of $β$ has cycles (The closure of a geometric braid); for $σ_{1}^{m} \in B_{2}$ the permutation is the transposition $(1 2)$ for odd $m$ and the identity for even $m$ , and the braid group is presented by the Artin generators (The braid group by Artin presentation).

Proof technique: direct substitution into the determinant formula.

1.1A2

The one-dimensional representation. By [A2] the image of $σ_{1}^{m}$ is the $1 \times 1$ matrix $(- t)^{m}$ , so $\det (I_{1} - {\overset{ˉ}{ρ}}_{2} (σ_{1}^{m})) = 1 - (- t)^{m}$ is the scalar displayed.

2.1A1step 1.1

The formula. Substituting step 1.1 into [A1] with $n = 2$ gives $Δ_{\hat{σ_{1}^{m}}} (t) ≐ (1 - t) (1 - (- t)^{m}) / (1 - t^{2})$ , the displayed formula.

3.1A3step 2.1algebra

The odd cases. For $m = 1$ : $(1 - t) (1 + t) / (1 - t^{2}) = 1$ , the unknot value. For $m = 3$ : $(1 - t) (1 + t^{3}) / (1 - t^{2}) = (1 + t^{3}) / (1 + t) = t^{2} - t + 1$ , the trefoil value. For $m = - 3$ : $(1 - t) (1 + t^{- 3}) / (1 - t^{2}) = (1 + t^{- 3}) / (1 + t) = t^{- 3} (t^{2} - t + 1)$ , which differs from $t^{- 2} - t^{- 1} + 1 = t^{- 2} (t^{2} - t + 1)$ by the unit $t^{- 1}$ , the mirror trefoil value; by [A3] these three closures are knots ( $m$ odd).

3.2A1A3step 2.1algebra

The even cases. For even $m$ the permutation of $σ_{1}^{m}$ is the identity, so by [A3] the closure has two components; the same display gives $Δ ≐ (1 - t) (1 - t^{m}) / (1 - t^{2})$ , and for $m = 2$ this is $(1 - t) (1 - t^{2}) / (1 - t^{2}) = 1 - t$ , the one-variable Alexander polynomial of the Hopf link in the convention of [A1]. The formula with the factor $(1 - t)$ is the classical formula for knots and links, so no separate knot hypothesis is needed for the value; the distinction is only that for even $m$ the closure is not a knot.

4.1A1A3step 3.2∎

The two-component value. The Hopf link is the closure of $σ_{1}^{2}$ , whose two components correspond to the two cycles of the identity permutation of $B_{2}$ by [A3]; its value $1 - t$ is the unit multiple $\pm t^{k}$ representative of the Alexander polynomial of the Hopf link in the normalization of [A1]. This completes the verification of the displayed values.

Remarks

The values $t^{2} - t + 1$ (trefoil) and $t^{- 2} - t^{- 1} + 1$ (mirror trefoil) are exchanged by $t \mapsto t^{- 1}$ , as the Alexander polynomial of mirror links requires.
The example is the smallest case of the determinant formula; the general $n$ case is The Burau determinant recovers the Alexander polynomial of a closed braid, and the Hopf-link value $1 - t$ agrees with the direct computation of the zeroth elementary ideal of its total-linking Alexander module.

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Sources

Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.2 equation (15) (printed p. 47) and Example 4.1 (printed p. 49) (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)