The admissible Wada quotient of a deficiency-one exterior spine

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