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The Burau determinant formula for a closed braid and its axis

Statement

Assume the Axiom of Choice (The Axiom of Choice) for the Alexander-module convention. Let $n \geq 2$ , let $β \in B_{n}$ with closure $\hat{β} \subset S^{3}$ and braid axis $A \subset S^{3}$ , so that $\hat{β} \cup A$ is an oriented link (The closure of a geometric braid); let ${\overline{B}}_{β} (t_{1}, \dots, t_{n})$ be the coloured reduced Burau matrix of The coloured reduced Burau matrix and put $B_{β} (t) : = {\overline{B}}_{β} (t, \dots, t)$ for its equal-label specialisation. Then:

(1) [Morton] the multivariable Alexander invariant of $\hat{β} \cup A$ satisfies $Δ_{\hat{β} \cup A} (t_{1}, \dots, t_{n}, x) ≐ \det (I - x {\overline{B}}_{β} (t_{1}, \dots, t_{n}))$ with the identifications $t_{π (j)} = t_{j}$ forced by the permutation $π$ of $β$ , where $x$ is the axis variable and $≐$ means equality up to multiplication by a unit of the Laurent ring $Z [t_{1}^{\pm 1}, \dots, t_{n}^{\pm 1}, x^{\pm 1}]$ ;

(2) [deletion of the axis] with the same identifications $t_{π (j)} = t_{j}$ , the Torres--Fox deletion of the axis gives the multivariable invariant of the closed braid, $D_{\hat{β}}^{m v} (t_{1}, \dots, t_{n}) ≐ \frac{\det (I - {\overline{B}}_{β} (t_{1}, \dots, t_{n}))}{1 - t_{1} t_{2} \dots t_{n}},$ and in the one-variable specialisation $t_{1} = \dots = t_{n} = t$ the one-variable Alexander polynomial of The Alexander polynomial from the zeroth elementary ideal satisfies $Δ_{\hat{β}} (t) ≐ \frac{(1 - t) \det (I - B_{β} (t))}{1 - t^{n}};$ Here $D^{m v}$ is Morton's multivariable invariant, a fraction for a knot. Its equal-label specialization satisfies $D_{\hat{β}}^{m v} (t, \dots, t) ≐ Δ_{\hat{β}} (t) / (1 - t)$ for any number of components; when there is more than one component it differs from the library's $D_{\hat{β}} = Δ_{\hat{β}}$ by the factor $1 - t$ . For a knot ( $π$ an $n$ -cycle) the polynomial equals $\det (I - B_{β} (t)) / (1 + t + \dots + t^{n - 1})$ up to units, and for a link with $k$ components the identifications leave one variable per cycle and the same formulas hold, with the library's Alexander invariant $D_{\hat{β}} = Δ_{\hat{β}} / (1 - t)$ for a knot and $D_{\hat{β}} = Δ_{\hat{β}}$ for $k > 1$ . All formulas are stated up to multiplication by a unit $\pm t^{m}$ (and $\pm t_{i}^{m}$ in the multivariable case) of the corresponding Laurent ring.

Facts & Assumptions

Given: AC and an integer $n \geq 2$ , a braid $β \in B_{n}$ with closure $\hat{β}$ and braid axis $A$ , the coloured reduced Burau matrix ${\overline{B}}_{β} (t_{1}, \dots, t_{n})$ , and the equal-label specialisation $B_{β} (t)$ . AC is inherited from the Alexander module; the finite Fox-determinant manipulations use no further choice.

[F1]

Literature input: diagram presentation. Use the bottom meridians $u_{1}, \dots, u_{n}$ of Morton's Figure 2 and their reverse partial products $g_{0} = 1$ , $g_{i} = u_{i} \dots u_{1}$ . For a crossing, let $Γ_{i}$ send $u_{i}$ to $u_{i + 1}$ and $u_{i + 1}$ to $u_{i + 1} u_{i} u_{i + 1}^{- 1}$ , fixing the other meridians. For the word $β = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{l}}^{ε_{l}}$ read from the top, successive substitutions express the top meridians in the bottom generators by $T_{β} = Γ_{i_{l}}^{ε_{l}} \circ \dots \circ Γ_{i_{1}}^{ε_{1}}$ . Gluing the two disk slices gives the complement presentation with generators $g_{1}, \dots, g_{n}, c$ and relations $T_{β} (g_{i}) = c^{- 1} g_{i} c$ , where $c$ is the axis meridian; $φ (u_{j}) = t_{j}$ and $φ (c) = x$ . This diagram presentation is the quoted topological input of Morton's proof of Theorem 1, printed pp. 4–6 (van Kampen, Seifert–van Kampen identifies the fundamental group with a group pushout). The substitutions here use moving disk slices; they are not the ordinary-composition automorphism of Artin automorphisms of the free group. No equality between those two word actions is assumed.

[F2]

Fox calculus. For a free basis $g_{1}, \dots, g_{n}$ , $\partial g_{k} / \partial g_{j} = δ_{k j}$ , $\partial (v w) / \partial g_{j} = \partial v / \partial g_{j} + v \partial w / \partial g_{j}$ , and $\partial (v^{- 1}) / \partial g_{j} = - v^{- 1} \partial v / \partial g_{j}$ . Writing $J (U)_{i j} = \partial U (g_{i}) / \partial g_{j}$ , ordinary function composition satisfies $J (U \circ V) = U (J (V)) J (U)$ , with $U$ applied entrywise to group-ring coefficients. These are the free-derivative rules used in Morton's proof, printed pp. 5–6. They give a product in word order for the successive-substitution action of [F1], not for the library's ordinary Artin word action.

[F3]

The deficiency-one Fox rule of The deficiency-one Fox calculus rule for the Alexander invariant: deleting the column of a generator $c$ with $φ (c) \neq 1$ and dividing the determinant of the remaining square matrix by $1 - φ (c)$ gives the evaluation of the Alexander invariant, up to a unit of $Z [H]$ .

[F4]

The coloured reduced Burau matrix of The coloured reduced Burau matrix is the matrix product of the ${\overline{C}}_{i} (a_{r})^{\pm 1}$ along an Artin word, with $a_{r}$ the label of the undercrossing string at crossing $r$ ; each factor is invertible with determinant $- a$ if the label is $a$ . At equal labels $t_{1} = \dots = t_{n} = t$ the specialisation is the standard reduced Burau matrix of The reduced Burau representation up to the fixed basis change of the coloured matrix; in particular the characteristic polynomials agree (Morton, Remark (1)).

[F5]

The one-variable Alexander polynomial $Δ_{L}$ and the Alexander invariant $D_{L} = Δ_{L}$ for a link with more than one component, $D_{L} = Δ_{L} / (1 - t)$ for a knot, of The Alexander polynomial from the zeroth elementary ideal, defined from the Alexander module of The one-variable Alexander module of an oriented link; the one-variable module is the cover classified by the total linking homomorphism.

[F6]

Literature input (quoted). Torres--Fox deletion (Morton, Remark (2), printed p. 3): for a link $L \cup C$ with meridian of $C$ replaced by $1$ , $D_{L} (t) = Δ_{L \cup C} (t, 1) / (1 - φ (c))$ , where $φ (c)$ is the element represented by $C$ in the complement of $L$ ; for the axis $C = A$ one has $φ (A) = t_{1} t_{2} \dots t_{n}$ (Conway, Theorem 3.15 and its proof, where the same deletion is computed through the twisted chain complex).

[F7]

Literature input (quoted). Birman--Brendle, section 4.2 equation (15): for the closure $b (X)$ of a braid $X \in B_{n}$ the Alexander polynomial satisfies $Δ_{b (X)} (t) = \det (\overset{ˉ}{ρ} (X) - I_{n - 1}) / (1 + t + \dots + t^{n - 1})$ up to the usual unit, where $\overset{ˉ}{ρ}$ is the reduced Burau representation; by Morton's Remark (1) the equal-label coloured matrix $B_{β} (t)$ is a matrix of that representation, so the same display reads $Δ_{\hat{β}} (t) ≐ (1 - t) \det (I - B_{β} (t)) / (1 - t^{n})$ up to sign. This one-variable normalization is the classical formula for knots and links and is quoted here; the identity between $B_{β} (t)$ and the matrix of the reduced Burau representation is verified in the next proposition on this page.

Proof

1.1F1givenalgebra

The diagram basis and elementary substitutions. The reverse partial products of [F1] are a free basis, since $u_{i} = g_{i} g_{i - 1}^{- 1}$ . Direct substitution gives $Γ_{i} (g_{i}) = g_{i + 1} g_{i}^{- 1} g_{i - 1}$ and $Γ_{i}^{- 1} (g_{i}) = g_{i - 1} g_{i}^{- 1} g_{i + 1}$ ; every other $g_{j}$ , including $g_{n}$ , is fixed. Thus [F1] supplies a deficiency-one presentation of the closed braid and axis, with $n + 1$ generators and $n$ relations. Deleting the column of $c$ is admissible because $φ (c) = x \neq 1$ .

2.1F1F2F4step 1.1algebra

The Jacobian product with transported labels. Put $Γ_{r} = Γ_{i_{r}}^{ε_{r}}$ and $S_{r} = Γ_{l} \circ \dots \circ Γ_{r}$ , with $S_{l + 1} = i d$ . Since $S_{r} = S_{r + 1} \circ Γ_{r}$ , [F2] gives $φ (J (S_{r})) = φ (S_{r + 1} (J (Γ_{r}))) φ (J (S_{r + 1}))$ . For a positive crossing, step 1.1 and the product rule give the exceptional row $(g_{i + 1} g_{i}^{- 1}, - g_{i + 1} g_{i}^{- 1}, 1)$ , truncated at $i = 1$ ; for a negative crossing the row is $(1, - g_{i - 1} g_{i}^{- 1}, g_{i - 1} g_{i}^{- 1})$ . The suffix $S_{r + 1}$ expresses the meridians immediately below crossing $r$ in the bottom generators. Hence the positive coefficient is $φ (S_{r + 1} (u_{i + 1})) = a_{r}$ , while the negative coefficient is $φ (S_{r + 1} (u_{i}))^{- 1} = a_{r}^{- 1}$ : these are precisely the undercrossing labels of [F4]. Their reduced blocks are ${\overline{C}}_{i_{r}} (a_{r})^{ε_{r}}$ . Iterating the displayed recurrence therefore gives $φ (J (T_{β})) = {\tilde{B}}_{β} = (\begin{array}{cc} {\overline{B}}_{β} & v \\ 0 & 1 \end{array})$ , with the leading factors in the defined word order. This calculation concerns $T_{β}$ of [F1].

3.1F1F2step 2.1algebra

The relation matrix. Differentiate $T_{β} (g_{i}) - c^{- 1} g_{i} c$ with respect to the $g_{j}$ . The second term evaluates to $x^{- 1} δ_{i j}$ , so deleting the column of $c$ leaves ${\tilde{B}}_{β} - x^{- 1} I_{n}$ . Its block form in step 2.1 gives $\det ({\tilde{B}}_{β} - x^{- 1} I_{n}) = (1 - x^{- 1}) \det ({\overline{B}}_{β} - x^{- 1} I_{n - 1})$ .

4.1F3step 1.1step 2.1step 3.1algebra

The Fox rule and the characteristic polynomial. Apply [F3] with the deleted generator $c$ and the divisor $1 - φ (c) = 1 - x$ , which cancels the explicit factor $1 - x^{- 1}$ up to the unit $- x^{- 1}$ of step 3.1: $Δ_{\hat{β} \cup A} ≐ - x^{- 1} \det ({\overline{B}}_{β} - x^{- 1} I_{n - 1}) ≐ \det (I - x {\overline{B}}_{β})$ , since $\det (x {\overline{B}}_{β} - I_{n - 1}) = (- 1)^{n - 1} \det (I - x {\overline{B}}_{β})$ and $x$ is a unit. The variable identifications $t_{π (j)} = t_{j}$ are those of the closed braid: strings joined at the top and bottom carry the same meridian. This proves (1).

5.1F4F6F7step 4.1algebra

Deletion of the axis. Put $x = 1$ and apply the Torres--Fox deletion of [F6] to the pair $(\hat{β}, A)$ with $φ (A) = t_{1} t_{2} \dots t_{n}$ and the identifications $t_{π (j)} = t_{j}$ ; part (1) at $x = 1$ gives the multivariable identity $D_{\hat{β}}^{m v} (t_{1}, \dots, t_{n}) ≐ \det (I - {\overline{B}}_{β} (t_{1}, \dots, t_{n})) / (1 - t_{1} t_{2} \dots t_{n})$ , the first display of (2). In the one-variable specialisation $t_{1} = \dots = t_{n} = t$ the denominator becomes $1 - t^{n}$ and the coloured matrix becomes $B_{β} (t)$ ; the one-variable normalization $Δ_{\hat{β}} (t) ≐ (1 - t) \det (I - B_{β} (t)) / (1 - t^{n})$ is the quoted classical formula [F7], so the equal-label multivariable invariant is $Δ_{\hat{β}} (t) / (1 - t)$ , rather than the library's multi-component normalization $D = Δ$ .

6.1F5F7step 5.1algebra∎

Knot and multi-component normalisations. Suppose first that $π$ is an $n$ -cycle, so that the closure is a knot. By [F5] the knot normalisation is $D = Δ / (1 - t)$ , and using $1 - t^{n} = (1 - t) (1 + t + \dots + t^{n - 1})$ gives $Δ_{\hat{β}} (t) ≐ \det (I - B_{β} (t)) / (1 + t + \dots + t^{n - 1})$ , equivalently $D_{\hat{β}} (t) ≐ \det (I - B_{β} (t)) / (1 - t^{n})$ up to units. If instead $π$ has $k$ disjoint cycles, the identifications $t_{π (j)} = t_{j}$ leave one variable per cycle, and the multivariable identity of step 5.1 is a $k$ -variable statement; the one-variable formula of step 5.1 is the classical formula [F7] and requires no knot hypothesis, so it computes $Δ_{\hat{β}}$ for a link as well, with $D_{\hat{β}} = Δ_{\hat{β}}$ for $k > 1$ by [F5]. Thus for $k > 1$ the latter is $(1 - t)$ times the equal-label specialization of $D^{m v}$ , as explicitly stated; the same determinant formula for the polynomial retains its factor $1 - t$ . This proves the displayed normalisations of (2).

Remarks

The unit ambiguity in (1) includes a power of $x$ ; the displayed form $\det (I - x {\overline{B}}_{β})$ is the normalization of Morton's Theorem 1.
The equal-label matrix $B_{β} (t)$ agrees with the matrix of the reduced Burau representation of The reduced Burau representation after the fixed basis change of [F4]; this is what The Burau determinant recovers the Alexander polynomial of a closed braid uses to rewrite the determinant for the representation-theoretic object.
The Fox rule [F3] is the quoted literature input of The deficiency-one Fox calculus rule for the Alexander invariant; steps 1.1–1.3 use Morton's diagram presentation and the explicitly ordered Fox chain rule, and the deletion of step 3.1 is Morton's Remark (2) and Conway's Theorem 3.15.

Depends on

Used by

The Burau determinant recovers the Alexander polynomial of a closed braid Proposition

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Sources

H. R. Morton, The multivariable Alexander polynomial for a closed braid, arXiv:math/9803138, section 2.1 and Theorem 1 with its complete proof (printed pp. 2-6) (standard reference, not scraped)
Anthony Conway, Burau maps and twisted Alexander polynomials, arXiv:1510.06678, Theorem 3.15 with its proof (printed pp. 16-17) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.2 equation (15) (printed p. 47) (standard reference, not scraped)