Seifert pairing and matrix of an oriented link

Assume full AC. Let $L=K_1\cup \cdots \cup K_r\subset S^3$ be a nonempty smooth oriented link, let $F$ be a connected compact smooth oriented embedded surface whose oriented boundary is $L$, and let $g$ be its genus. The compact-surface normal form gives $H_1(F;\mathbb{Z})\cong {\mathbb{Z}}^{2g+r-1}$. Orient the normal line of $F$ by requiring $(n,u,v)$ to be a positive ambient frame whenever $(u,v)$ is a positive tangent frame of $F$. For an ordered integral basis $a_1,\dots ,a_m$, with $m=2g+r-1$, choose circle-cycle representatives on $F$ and let $a_i^{+}$ be a sufficiently small copy in the positive normal direction. For disjoint oriented embedded circles $C,D\subset S^3$, define $\mathrm{lk}(C,D)$ by $[C]=\mathrm{lk}(C,D)[{\mu }_D]$ in $H_1(S^3\setminus D;\mathbb{Z})$, where ${\mu }_D$ is the positive meridian whose oriented normal disk follows the positive tangent to $D$ in the ambient orientation; extend linking bilinearly to finite sums of such circles. Define the positive-push Seifert matrix by $V_{ij}=\mathrm{lk}(a_i^{+},a_j)$. The link-surface pairing lemma proves this matrix is independent of the chosen cycle representatives and push-off level. The surface and basis remain part of the data; this definition asserts no link-only matrix invariance.

Smooth oriented links in the oriented three-sphere

The rank of first homology for a compact oriented surface with boundary

Alexander-duality meridian coordinates for an oriented link complement

Well-definedness of the positive-push pairing for link surfaces

Full choice for the Alexander-duality linking convention