Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 112 / 2 / d / Solution

For the supplied Seifert matrix $A$,and the symmetric form $Q=A+A^T$ has signature $-2$. ThusSince the Levine-Tristram signature is an additive homomorphism on the algebraic concordance group, $K$ has infinite algebraic-concordance order.

Over ${\mathbb{Q}}_3$, reduction of the Alexander polynomial givesThe factors are coprime, nonsymmetric, and exchanged by reciprocity. Hensel lifting therefore decomposes the local isometric structure into a reciprocal pair, which is metabolic. Its class in ${\mathcal{W}}_{{\mathbb{Q}}_3}$ is zero and in particular does not have order four.

For $p=11$, diagonalization givesThe second residue at $11$ is the one-dimensional formBecause $11\equiv 3(\mathrm{mod}4)$, $W({\mathbb{F}}_{11})\cong \mathbb{Z}/4$ and this one-dimensional form is a generator. The P-adic algebraic-concordance obstruction therefore has exact order four, so the image of $K$ in ${\mathcal{W}}_{{\mathbb{Q}}_{11}}$ has order four.