Twisted Alexander polynomials and Reidemeister torsion

Assume the Axiom of Choice as in the maximal-abelian-cover convention. Let $L=K_1\cup \cdots \cup K_r\subset S^3$ be a nonempty oriented smooth link, with $r\ge 1$, compact exterior $E_L$ formed by deleting the interiors of pairwise-disjoint chosen closed normal-disk tubes, and basepoint $b\in E_L$. For a knot, $r=1$ and $E_L=E_K$. The full meridian abelianization group is $H_L=H_1(E_L;\mathbb{Z})\cong {\mathbb{Z}}^r$, with the positive normal-disk meridians as its ordered basis. Choose a finite-rank free abelian group $H$ of positive rank and an epimorphism $\chi :H_L\to H$; write $\psi =\chi \circ \mathrm{a}\mathrm{b}:G\to H$ for the induced map from $G={\pi }_1(E_L,b)$. The identity choice $H=H_L$ is the full multivariable deck map; a coloring may identify its variables only through the explicitly stated map $\chi$. Let $\mathbb{F}$ be a field, let $d\ge 1$, and let $\rho :G\to {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$ be a representation. Put $\mathrm{\Lambda }=\mathbb{F}[H]$, the commutative Laurent-polynomial ring with monomial basis $H$, and $Q=\mathrm{Frac}(\mathrm{\Lambda })$. On row vectors define the right $\mathbb{F}[G]$ action by $v\cdot g=v\rho (g)\psi (g)$, where $\psi (g)$ is central multiplication by its monomial in $\mathrm{\Lambda }$. For a chosen finite based CW structure $X$ on $E_L$, orient and order its cells and choose one lift of each cell to the universal cover $\tilde{X}$. The twisted cellular complex is

$C_{\ast }^{\rho ,\psi }(X;{\mathrm{\Lambda }}^d)={\mathrm{\Lambda }}^d{\otimes }_{\mathbb{F}[G]}{C}_{\ast }(\tilde{X};\mathbb{F}),$

and its fraction-field complex is $C_{\ast }^{\rho ,\psi }(X;Q^d)=Q{\otimes }_{\mathrm{\Lambda }}{C}_{\ast }^{\rho ,\psi }(X;{\mathrm{\Lambda }}^d)$. The lifted cell bases and the standard coordinate basis of ${\mathbb{F}}^d$ give the based $Q$-vector-space bases. If this fraction-field complex is not acyclic, define its twisted torsion to be $0$. If it is acyclic, choose in each degree a basis ${\mathbf{b}}_i$ of $B_i=\mathrm{im}{\mathrm{\partial }}_{i+1}$ and lifts ${\tilde{\mathbf{b}}}_{i-1}$ of ${\mathbf{b}}_{i-1}$ under ${\mathrm{\partial }}_i$; with ${\mathbf{c}}_i$ the based cellular basis, set

$\tau (C_{\ast },\mathbf{c})={\prod }_i[{\mathbf{b}}_i,{\tilde{\mathbf{b}}}_{i-1}\mathrm{/}{\mathbf{c}}_i{]}^{(-1{)}^{i+1}}\in Q^{\times },$

where brackets denote the determinant of the change-of-basis matrix. The twisted Reidemeister torsion is this value modulo the subgroup ${\mathcal{U}}_{\rho ,\psi }=\{\pm \det (\rho (g))\psi (g{)}^d:g\in G\}\subset Q^{\times }$. This exact determinant image is contained in the commonly used coarser unit subgroup $\{\pm ah:a\in \det (\rho (G)),h\in H\}$. The choice-free statement uses the full abelianization chain model first and specializes it only by the declared coefficient map $\chi$.

Full abelian cover and positive-meridian variables

Conway, §§2.2–2.4 and 3.3

Kirk and Livingston, §§2.1–3

Ranicki, Definitions 4 and 6

Assume the Axiom of Choice convention of the twisted-chain definition. Let $L$ be a nonempty oriented link, let $E_L$ be its chosen compact exterior, and let $X$ be a finite based two-dimensional CW complex with one $0$-cell, $n\ge 2$ oriented $1$-cells $x_1,\dots ,x_n$, and $n-1$ oriented $2$-cells attached by based words $r_1,\dots ,r_{n-1}$. The supplied data include a based cellular spine certificate: a finite sequence of elementary cellular expansions and collapses identifying the simple-homotopy type of $X$ with the selected finite CW structure on $E_L$. Thus an abstract group isomorphism ${\pi }_1(X)\cong {\pi }_1(E_L)$ alone is not a spine certificate. Let $G={\pi }_1(E_L)$, let $\rho :G\to {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$ and $\psi :G\twoheadrightarrow H$ be the representation and declared free-abelian deck map, and put $Q=\mathrm{Frac}(\mathbb{F}[H])$. The associated deficiency-one presentation is $⟨x_1,\dots ,x_n\mid r_1,\dots ,r_{n-1}⟩$ through the certified spine map. For $w\in F_n$, define Fox derivatives recursively by $\mathrm{\partial }{x}_k\mathrm{/}\mathrm{\partial }{x}_j={\delta }_{kj}$, $\mathrm{\partial }(uv)\mathrm{/}\mathrm{\partial }{x}_j=\mathrm{\partial }u\mathrm{/}\mathrm{\partial }{x}_j+u\mathrm{\partial }v\mathrm{/}\mathrm{\partial }{x}_j$, and $\mathrm{\partial }{x}_k^{-1}\mathrm{/}\mathrm{\partial }{x}_j=-x_k^{-1}{\delta }_{kj}$. Extend them linearly to the group ring and let $\mathrm{\Phi }:\mathbb{F}[F_n]\to M_d(Q)$ be the ring homomorphism induced by $x_k\mapsto \rho (x_k)\psi (x_k)$. Form the block Fox matrix $J=(\mathrm{\Phi }(\mathrm{\partial }{r}_i\mathrm{/}\mathrm{\partial }{x}_k){)}_{i,k}$, of size $(n-1)d$ by $nd$. For a chosen column index $j$, let $J_{\widehat{j}}$ be obtained by deleting its entire $d$-column block and let $D_j=\det (\rho (x_j)\psi (x_j)-I_d)$. The Wada quotient for this column is defined only when $D_j\mathrm{\ne }0$, and then is $W_j=\det (J_{\widehat{j}})\mathrm{/}{D}_j\in Q$; a zero denominator makes that column inadmissible even when another column is valid. No abstract deficiency-one presentation is assigned exterior torsion unless its presentation complex is the stated based spine.

Twisted coefficient complex and torsion convention

Kirk and Livingston, §4

Conway, §§2.4 and 3.3

Ranicki, Definition 6 and Theorem 15

Assume the Axiom of Choice convention of the twisted-chain definition. Let $X$ be a connected finite based two-dimensional CW complex with one $0$-cell, $n\ge 2$ $1$-cells and $n-1$ $2$-cells, so $\chi (X)=0$. Let $G={\pi }_1(X)$, let $\psi :G\twoheadrightarrow H\cong {\mathbb{Z}}^m$ be an epimorphism with $m\ge 1$, let $\rho :G\to {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$ be a representation over a field $\mathbb{F}$, set $\mathrm{\Lambda }=\mathbb{F}[H]$ and $Q=\mathrm{Frac}(\mathrm{\Lambda })$, and give ${\mathrm{\Lambda }}^d$ the right $\mathbb{F}[G]$-action $v\cdot g=v\rho (g)\psi (g)$, with $\psi (g)$ acting by its central monomial. Form

$C_{\ast }^{\rho ,\psi }(X;{\mathrm{\Lambda }}^d)={\mathrm{\Lambda }}^d{\otimes }_{\mathbb{F}[G]}{C}_{\ast }(\tilde{X};\mathbb{F}),$

where $\tilde{X}$ is the universal cover, and set $C_{\ast }^{\rho ,\psi }(X;Q^d)=Q{\otimes }_{\mathrm{\Lambda }}{C}_{\ast }^{\rho ,\psi }(X;{\mathrm{\Lambda }}^d)$. If every element of $H_1(C_{\ast }^{\rho ,\psi }(X;{\mathrm{\Lambda }}^d))$ is annihilated by some nonzero element of $\mathrm{\Lambda }$, then $C_{\ast }^{\rho ,\psi }(X;Q^d)$ is acyclic. No assumption that its degree-two homology vanishes before localization is required.

Twisted coefficient complex and deck action

Conway, Lemma 2.6

Kirk and Livingston, §§3–4

Ranicki, Example 17

Assume the Axiom of Choice convention of the twisted-chain definition. Let $X$ be a finite based CW model of a link exterior, let $G={\pi }_1(X)$, let $\alpha (g)=\rho (g)\psi (g)\in {\mathrm{G}\mathrm{L}}_d(Q)$ be its declared twisted coefficient representation, and suppose the based fraction-field complex is acyclic. Let $\mathcal{U}=\{\pm \det (\alpha (g)):g\in G\}$. Changing the lift or orientation of any cell multiplies the based Reidemeister torsion by an element of $\mathcal{U}$, so its class in $Q^{\times }\mathrm{/}\mathcal{U}$ is unchanged. An elementary simple expansion or collapse, meaning addition or deletion of a pair of adjacent-dimensional cells whose lifted incidence block is $\pm g$ for some $g\in G$, has the same property.

Based chain and torsion convention

Ranicki, basis changes and Theorem 15

Conway, §§2.3–2.4

Kirk and Livingston, §3

Let $X$ be the based presentation $2$-complex with one vertex, oriented $1$-cells $x_1,\dots ,x_n$, and oriented $2$-cells attached along words $r_1,\dots ,r_m$ in the free group $F_n$. Let $G=⟨x_1,\dots ,x_n\mid r_1,\dots ,r_m⟩$ be its presented fundamental group, let $\rho :G\to {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$ be a representation, and let $\psi :G\to H$ be the declared free-abelian deck map. For $\mathrm{\Phi }:\mathbb{F}[F_n]\to M_d(Q)$ induced by $x_j\mapsto \rho (x_j)\psi (x_j)$, the twisted differential ${\mathrm{\partial }}_1$ has blocks $\rho (x_j)\psi (x_j)-I_d$, and the $i$th block row of ${\mathrm{\partial }}_2$ is

$(\mathrm{\Phi }\left(\frac{\mathrm{\partial }{r}_i}{\mathrm{\partial }{x}_1}\right),\dots ,\mathrm{\Phi }\left(\frac{\mathrm{\partial }{r}_i}{\mathrm{\partial }{x}_n}\right)),$

where coefficient vectors are written as rows and differentials act on the right: the matrix of ${\mathrm{\partial }}_1$ is the stacked column of blocks $\rho (x_j)\psi (x_j)-I_d$, while the matrix of ${\mathrm{\partial }}_2$ is the $md$-by-$nd$ block matrix whose $i$th row is the displayed Fox row. The Fox derivatives use the product and inverse recursions in the Wada definition. Thus the evaluated Fox-Jacobian blocks are exactly the cellular boundary blocks of the based twisted complex of this presentation $2$-complex.

Twisted chain convention

Fox derivative and Wada matrix conventions

Conway, §§2.4 and 3.1

Kirk and Livingston, §4

Ranicki, pp. 3–5

Assume the Axiom of Choice convention of the twisted-chain definition. Let $L$ be a nonempty oriented link and let $X$ be a certified finite based two-dimensional spine of its compact exterior with one $0$-cell, $n\ge 2$ $1$-cells and $n-1$ $2$-cells. Let $\psi :{\pi }_1(X)\twoheadrightarrow H\cong {\mathbb{Z}}^m$ be an epimorphism with $m\ge 1$, let $\rho :{\pi }_1(X)\to {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$ be a representation over a field, and form the twisted chain complex over $\mathrm{\Lambda }=\mathbb{F}[H]$ and its localization over $Q=\mathrm{Frac}(\mathrm{\Lambda })$. If its degree-one homology over $\mathrm{\Lambda }$ is a torsion module, then its fraction-field complex is acyclic.

Twisted exterior chain complex

General localized acyclicity lemma

Conway, Lemma 2.6

Kirk and Livingston, §3

Assume the Axiom of Choice convention of the twisted-chain definition. Let $X$ be a certified based deficiency-one exterior spine with one $0$-cell, $n\ge 2$ $1$-cells and $n-1$ $2$-cells, let $\psi :{\pi }_1(X)\twoheadrightarrow H\cong {\mathbb{Z}}^m$ have $m\ge 1$, let $\rho :{\pi }_1(X)\to {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$, and let $H_1(C_{\ast }^{\rho ,\psi }(X;\mathbb{F}[H{]}^d))$ be torsion over $\mathbb{F}[H]$. Form the Fox block matrix $J$ and a chosen Wada column quotient $W_j=\det (J_{\widehat{j}})\mathrm{/}{D}_j$ with $D_j=\det (\rho (x_j)\psi (x_j)-I_d)\mathrm{\ne }0$. Then the fraction-field complex is acyclic, $\det (J_{\widehat{j}})\mathrm{\ne }0$, and $W_j$ equals its based twisted Reidemeister torsion in $Q^{\times }\mathrm{/}{\mathcal{U}}_{\rho ,\psi }$. Every admissible deleted column gives the same class. This conclusion applies only to the specified cellular spine; an abstract presentation of the same group is insufficient.

Wada quotient and denominator

Fox Jacobian as the cellular differential

Localized acyclicity

Torsion unit ambiguity

Kirk and Livingston, Theorem 4.1 and Lemma 4.2

Conway, §§2.3–2.4

Ranicki, Theorem 15

Inputs. An integer $n\ge 2$; a finite Artin word $\beta \in B_n$ in the positive-half-twist convention where ${\sigma }_i$ sends the $i$th meridian $x_i$ to $x_i{x}_{i+1}{x}_i^{-1}$ and $x_{i+1}$ to $x_i$; a surjective color map $c$ from the cycles of the endpoint permutation to $\{1,\dots ,\mu \}$, written on strands as $c_1,\dots ,c_n$ and constant around each cycle; a field $\mathbb{F}$ with encoded exact arithmetic and equality; a positive integer $d$ and invertible matrices $\rho (x_i)\in {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$; and compatible finite certificates of based deficiency-one $2$-spines for the closure exterior $X$ and the closure-with-axis exterior $Y$. The $X$ spine has generators $g_i=x_1\cdots x_i$ and relators $r_i=g_i^{-1}{\beta }_{\ast }(g_i)$ for $1\le i<n$. The $Y$ spine has generators $g_1,\dots ,g_n,x$ and relators $s_i={\beta }_{\ast }(g_i)(x^{-1}{g}_{i}x{)}^{-1}$ for $1\le i\le n$; $x$ is the meridian of the axis. The certificates include a based cell structure on $T$ with one vertex, $1$-cells $x,g_n$, and the commutator $2$-cell; the one-vertex, one-core-edge structure on $V=D^2\times S^1$; and compatible cell bases for $X=Y{\cup }_{T}V$. These spine and gluing certificates are geometric input and are not recognized by the procedure. Put $H=⟨t_1,\dots ,t_{\mu }⟩\cong {\mathbb{Z}}^{\mu }$, $\psi (x_i)=t_{c_i}$, $g_n=x_1\cdots x_n$, $\mathrm{\Lambda }=\mathbb{F}[H]$, and $Q=\mathrm{Frac}(\mathrm{\Lambda })$.

Instructions.

1. Read the supplied $X$ and $Y$ spine/presentation correspondences and compatible gluing as certified geometric input; check their finite cell counts and relator lists, but do not attempt to recognize whether the data describe exterior spines. Return not-applicable if either certificate is absent. Compute the induced free-group automorphism ${\beta }_{\ast }$ by substituting the displayed Artin rules in the declared word order. For the mapping-torus checks, set the axis meridian's representation matrix to $\rho (x)=I_d$. Return not-applicable if the coloring is not preserved on endpoints, if the representation fails any closure relator, or if it fails any mapping-torus relator $s_i={\beta }_{\ast }(g_i)(x^{-1}{g}_{i}x{)}^{-1}$. Equivalently the braid invariance checks $\rho ({\beta }_{\ast }(x_i))=\rho (x_i)$ must hold for all meridians.

2. Set $\psi (x)=1$ on the axis meridian. Verify the deck map on the relators $r_i$ and the axis relators $s_i$, and verify the colors are surjective, so $\psi$ descends to both spine groups and is an epimorphism. Compute the positive-meridian full abelianization basis first, then use the explicit color map $H_L\to H$; no unannounced equal-variable or total-linking specialization is performed.

3. Rewrite ${\beta }_{\ast }(g_i)$ in the free basis $g_1,\dots ,g_n$, using $x_i=g_{i-1}^{-1}{g}_i$ and $g_0=1$. Compute the block Fox matrix $B_{\beta }$ with $(i,j)$ block $\mathrm{\Phi }(\mathrm{\partial }{\beta }_{\ast }(g_i)\mathrm{/}\mathrm{\partial }{g}_j)$, using the product and inverse recursions of the Wada definition and $\mathrm{\Phi }(g)=\rho (g)\psi (g)$. The braid fixes the boundary word $g_n$, so its last block row is $(0,\dots ,0,I_d)$. Let ${\bar{B}}_{\rho }(\beta )$ be the upper-left $(n-1)d$-square block.

4. Compute $D=\det (\rho (g_n)\psi (g_n)-I_d)$. If $D=0$, return not-applicable: this designated deleted column is inadmissible, even if another column of the Fox matrix could be used.

5. Form the twisted cellular boundary matrices $d_1,d_2$ of the closure-exterior spine $X$ from the Fox lemma. Use exact Gaussian elimination over $Q$ to check the chain identity $d_1\circ d_2=0$ (the row-matrix product $J{A}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}=0$). If this identity fails, return not-applicable because the encoded data do not define a chain complex. If it holds but the ranks are not $\mathrm{rank}(d_1)=d$ and $\mathrm{rank}(d_2)=(n-1)d$, return $0$ under the declared torsion convention. Otherwise compute $N=\det ({\bar{B}}_{\rho }(\beta )-I_{(n-1)d})$ and return the rational function $N\mathrm{/}D$ as the twisted Burau determinant value, modulo the determinant/deck units of the based torsion convention.

The braid convention is shown for two adjacent downward-oriented strands. The left strand crosses over the right strand for ${\sigma }_i$; the gap records the underpass. The word, color map, and algebraic action above, rather than the diagram, determine the computation.

The displayed generator interchanges endpoint positions, with the original left strand passing over; the colors must follow the same permutation along each link component.

Twisted coefficient and torsion conventions

Admissible Fox quotient and denominator

Positive Artin action on punctured-disk meridians

Well-defined Artin representation of the braid group

Identification with the geometric braid action

Closed-braid closure and component cycles

Conway, §§2.1–2.4 and 3.1–3.3

Kirk and Livingston, §§2–4

Ranicki, Definitions 4 and 6

Assume the Axiom of Choice convention of the twisted-chain definition. Let $\beta \in B_n$ be a geometric braid with $n\ge 2$ strands, let $X$ be the compact exterior of its oriented closure, and let $Y$ be the compact exterior of the closure together with the braid axis. Suppose finite based deficiency-one two-dimensional spine certificates are supplied for $X$ and $Y$, with compatible cell bases for the filling decomposition $X=Y{\cup }_{T}V$, where $T$ is the boundary torus and $V\cong D^2\times S^1$ is the filled solid torus. The certificate for $X$ has generators $g_1,\dots ,g_n$, where $g_i=x_1\cdots x_i$ are cumulative positive meridians, and relators $r_i=g_i^{-1}{\beta }_{\ast }(g_i)$ for $1\le i<n$; the certificate for $Y$ has generators $g_1,\dots ,g_n,x$, where $x$ is the axis meridian, and relators $s_i={\beta }_{\ast }(g_i)(x^{-1}{g}_{i}x{)}^{-1}$ for $1\le i\le n$. The gluing certificate gives $T$ its one-vertex cell structure with $1$-cells $x,g_n$ and one commutator $2$-cell, and gives $V$ its one-vertex and one-core-edge structure, with compatible based cell bases. Assume these are actual compatible based spine and gluing certificates; an abstract group isomorphism or matching relator list alone is insufficient.

Let $H_L=H_1(X;\mathbb{Z})$ have its ordered positive-meridian basis, let $\chi :H_L\twoheadrightarrow H\cong {\mathbb{Z}}^m$ be the declared color map with $m\ge 1$, and write $\psi$ for the induced deck map. Let $\mathbb{F}$ be a field, let $\rho :{\pi }_1(X)\to {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$ be a representation, and use the coefficient systems on $Y,T,V$ pulled back or extended across the filling; thus $x$ acts as the identity and the core $g_n$ of $V$ acts by $M_n=\rho (g_n)\psi (g_n)$. Put $D=\det (M_n-I_d)$ and assume $D\mathrm{\ne }0$. Compute the reduced colored Burau block ${\bar{B}}_{\rho }(\beta )$ in the cumulative-meridian basis as in the finite procedure, and put $N=\det ({\bar{B}}_{\rho }(\beta )-I_{(n-1)d})$. For the declared convention that torsion is zero on a nonacyclic fraction-field complex, if the twisted complex of $X$ is acyclic then its torsion is represented by $N\mathrm{/}D$ modulo the exact determinant and deck-unit subgroup of the coefficient system. If the complex is nonacyclic then $N=0$ and $\tau (X)=N\mathrm{/}D=0$ in $Q$. The acyclic identity follows from the braid-axis filling product.

The diagram records only the supplied gluing certificate; it is schematic and is not a substitute for the cell or spine data.

Colored Burau block and boundary denominator conventions

Well-defined Artin representation of the braid group

Identification with the geometric braid action

Twisted coefficient and torsion conventions

Conway, Theorem 3.15 and proof

Kirk and Livingston, §§2–4

Ranicki, Definitions 4 and 6 and Theorem 15