Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 112 / 1 / a

A Seifert surface for an oriented knot $K$ is a compact connected oriented surface $F\subset S^3$ whose oriented boundary is $K$. For homology classes represented by oriented curves $x,y\subset F$, the Seifert form iswhere $x^{+}$ is the positive normal push-off. Choosing a basis of $H_1(F;\mathbb{Z})$ gives a Seifert matrix $A$.

For $\omega \in S^1\setminus \{1\}$, the Levine-Tristram signature isThe determinant of this Hermitian matrix vanishes away from $\omega =1$ exactly at the unit roots of the Alexander polynomial of a knot ${\Delta }_K$. Consequently the signature is locally constant on their complement.

For $\omega =e^{i\theta }$ near $1$,The real skew-symmetric unimodular matrix $A^T-A$ has standard symplectic blocks, so the Hermitian matrix $i(A^T-A)$ has its positive and negative eigenvalues in opposite pairs and has signature zero. Thus ${\sigma }_{\omega }(K)=0$ near $1$. If ${\Delta }_K$ has no unit roots, then $S^1\setminus \{1\}$ contains no singular point of the signature form and is connected, so local constancy gives ${\sigma }_{\omega }(K)=0$ everywhere. With the usual convention ${\sigma }_1(K)=0$, the signature vanishes identically.

Solved by gpt-5.6-sol high.