S-equivalence and Conway normalization

Assume full AC. For an even-order integral matrix $V$ that is the Seifert matrix of a knot, S-equivalence means the equivalence relation generated by integral unimodular congruences $V\mapsto P^{T}VP$ with $\det P=\pm 1$ and the following two enlargements, for column vectors $x,y\in {\mathbb{Z}}^n$ and $z\in \mathbb{Z}$:

$E_0(V;x,y,z)=\left(\begin{array}{ccc}V& x& 0\\ y^T& z& 0\\ 0^T& 1& 0\end{array}\right),E_1(V;x,y,z)=\left(\begin{array}{ccc}V& x& 0\\ y^T& z& 1\\ 0^T& 0& 0\end{array}\right),$ and their reverses. For an assignment $P$ of an integer polynomial in $z$ to each ambient-isotopy class of oriented links, call $P$ Conway-normalized when $P_{\mathrm{\varnothing }}(z)=0$ for the empty link, $P_U(z)=1$ for the unknot $U$, and $P_{L_{+}}(z)-P_{L_{-}}(z)=z{P}_{L_0}(z)$ whenever the three oriented links agree outside one crossing ball and inside it are respectively the positive crossing, negative crossing, and orientation-compatible smoothing; positive means ${\det }_P(v_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}},v_{\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}})>0$, where $v_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}},v_{\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}}$ are the projected oriented tangent vectors and $P$ is oriented so the viewing normal followed by a positive basis of $P$ is ambient-positive. The smoothing reconnects the four oriented half-arcs in the unique noncrossing way compatible with their directions. No existence of such an assignment is presupposed: the Seifert-determinant lemma constructs the unique assignment on nonempty links; extending it by ${\mathrm{\nabla }}_{\mathrm{\varnothing }}(z)=0$ gives the unique assignment on this full domain, denoted by ${\mathrm{\nabla }}_L(z)$. For a knot, let ${\mathrm{\Delta }}_K^{\mathrm{\#}}(t)$ be the unique associate of its Alexander polynomial satisfying ${\mathrm{\Delta }}_K^{\mathrm{\#}}(1)=1$ and ${\mathrm{\Delta }}_K^{\mathrm{\#}}(t^{-1})={\mathrm{\Delta }}_K^{\mathrm{\#}}(t)$; existence and uniqueness follow from the Alexander-normalization corollary. Set $z=t^{1\mathrm{/}2}-t^{-1\mathrm{/}2}$, with $t^{1\mathrm{/}2}$ a formal Laurent variable, then ${\mathrm{\Delta }}_K^{\mathrm{\#}}(t)={\mathrm{\nabla }}_K(z)$, as justified by the centered determinant below.

The two integral enlargement matrices and congruence convention

The Alexander polynomial from the zeroth elementary ideal

Existence and uniqueness of the symmetric normalization with value one at $t=1$

Existence and uniqueness of the Conway-normalized assignment ↗