Twisted chain complex and based Reidemeister torsion for link exteriors

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