The Seifert-matrix map in the infinite cyclic cover

Assume the Axiom of Choice. Let $K$ be a smooth oriented knot in the oriented standard three-sphere, and let $F$ be a connected compact smooth oriented embedded surface with boundary orientation equal to that of $K$. Let $g\ge 0$ satisfy $\chi (F)=1-2g$. Choose a small product tubular neighborhood $N(K)$ adapted to the inward conormal of $F$, put $E=S^3\setminus \mathrm{int}N(K)$ and $S=F\cap E$, and cut $E$ open along a product bicollar of $S$ to obtain the compact cut-open complement $Y$. Orient the normal line $n$ of $F$ by requiring $(n,u,v)$ to be positive in $S^3$ whenever $(u,v)$ is a positive tangent basis of $F$; name the face on the $+n$ side $S_{+}$ and the face on the $-n$ side $S_{-}$. Let $j:S\hookrightarrow F$ be the collar-removal inclusion and ${\iota }_{\pm }:S\to Y$ the two face inclusions; the inclusion induces an isomorphism on $H_1$, and set $i_{\pm }=({\iota }_{\pm }{)}_{\ast }\circ j_{\ast }^{-1}$. Form ${\tilde{E}}_F$ from copies $Y_n$ by gluing $S_{-}(Y_n)$ to $S_{+}(Y_{n+1})$, and let $t:Y_n\to Y_{n+1}$ be the deck shift. Because the boundary orientation of $F$ agrees with $K$, this shift is the positive-meridian shift. Let $a_1,\dots ,a_{2g}$ be an ordered integral basis of $H_1(F;\mathbb{Z})$ and let $V_{ij}=\mathrm{lk}(a_i^{+},a_j)$, where $a_i^{+}$ is the positive normal push-off. In the linking-dual basis given by $[\gamma ]\mapsto ([a]\mapsto \mathrm{lk}(\gamma ,a))$ under $H_1(Y;\mathbb{Z})\cong \mathrm{Hom}(H_1(F;\mathbb{Z}),\mathbb{Z})$, the map $i_{-}-t{i}_{+}$ is represented over $\mathrm{\Lambda }=\mathbb{Z}[t^{\pm 1}]$ by $V-t{V}^T$. Consequently $H_1({\tilde{E}}_F;\mathbb{Z})$ is the cokernel of $V-t{V}^T$.

The positive-push pairing and matrix entries

Symmetry and push-level invariance of linking

The cut-open cover, linking-dual identification, and Mayer–Vietoris map

The Laurent polynomial ring and deck-variable convention

The coordinate square records the map and its matrix. The vertical arrows send a surface class to its chosen basis coordinates and a cut-complement class to its linking evaluations on that basis; step 1.1 proves commutativity entry by entry.

diagram failed to render:
\begin{tikzpicture}[>=stealth,node distance=1.6cm]
  \node (h1) {$H_1(F)\otimes\Lambda$};
  \node[right=3.2cm of h1] (h2) {$H_1(Y)\otimes\Lambda$};
  \node[below=1.2cm of h1] (z1) {$\Lambda^{2g}$};
  \node[below=1.2cm of h2] (z2) {$\Lambda^{2g}$};
  \draw[->] (h1) -- node[above] {$i_- -t i_+$} (h2);
  \draw[->] (h1) -- node[left] {basis coordinates} (z1);
  \draw[->] (h2) -- node[right] {linking dual} (z2);
  \draw[->] (z1) -- node[below] {$V-tV^T$} (z2);
\end{tikzpicture}