Seifert matrix ($A$)

A Seifert matrix represents the Seifert form in an integral basis of $H_1(F)$. Its skew-symmetrization $A-A^T$ represents the intersection form of the surface and is unimodular.

Up to multiplication by a unit $\pm t^n$, the Alexander polynomial isIt satisfies ${\Delta }_K(t^{-1})\doteq {\Delta }_K(t)$ and ${\Delta }_K(1)=\pm 1$.

The knot determinant is $|{\Delta }_K(-1)|=|\det (A+A^T)|$. It is also the order of the first homology of the two-fold cover of $S^3$ branched over $K$.

For $\omega \in S^1\setminus \{1\}$, the Levine-Tristram signature is the signature of the Hermitian matrixIt is locally constant away from unit roots of the Alexander polynomial of a knot.