Levine–Tristram Hermitian form and signature

Assume full AC. Let $K$ be a smooth oriented knot in the oriented standard $3$-sphere, let $F\subset S^3$ be a connected compact smooth oriented embedded surface with oriented boundary $K$, let $V$ be the matrix of the Seifert pairing $V_{ij}=\mathrm{lk}(a_i^{+},a_j)$ in an ordered integral basis, and let $\omega \in S^1\setminus \{1\}$. Define the Levine–Tristram Hermitian matrix by $H_{\omega }(V)=(1-\omega )V+(1-\bar{\omega })V^T\mathrm{.}$ It is Hermitian because $V$ is real and $\bar{\omega }={\omega }^{-1}$. Its signature is the number of positive eigenvalues minus the number of negative eigenvalues, both counted with multiplicity and with zero eigenvalues omitted. When ${\mathrm{\Delta }}_K(\omega )\mathrm{\ne }0$, define ${\sigma }_{\omega }(K)$ to be this signature. At a root ${\omega }_0\mathrm{\ne }1$ of ${\mathrm{\Delta }}_K$, define ${\sigma }_{{\omega }_0}(K)$ to be the arithmetic mean of the two one-sided limits of ${\sigma }_{e^{i\theta }}(K)$ as $\theta$ approaches an argument of ${\omega }_0$ from below and above. The invariance and one-sided-existence lemmas justify treating these values as independent of the chosen surface and basis.

The Seifert matrix from positive push-offs and linking numbers

Singularity occurs exactly at Alexander roots

Surface and basis invariance ↗

Existence of one-sided limits at roots ↗