For the supplied <Seifert matrix> $A$,
$$
\Delta_K(t)=\det(tA-A^T)
=-t^4+3t^3-3t^2+3t-1,
$$
and the symmetric form $Q=A+A^T$ has signature $-2$. Thus
$$
\sigma_{-1}(K)=-2\ne0.
$$
Since the <Levine-Tristram signature> is an additive homomorphism on the <algebraic concordance of knots>[algebraic concordance group], $K$ has infinite algebraic-concordance order.

Over $\mathbb Q_3$, reduction of the Alexander polynomial gives
$$
-\Delta_K(t)\equiv(t^2+t-1)(t^2-t-1)\pmod3.
$$
The 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 gives
$$
Q\sim\left\langle-2,-\frac32,2,-\frac{11}{6}\right\rangle.
$$
The second residue at $11$ is the one-dimensional form
$$
\left\langle-\frac16\right\rangle
=\langle9\rangle=\langle1\rangle
\quad\text{in }W(\mathbb F_{11}).
$$
Because $11\equiv3\pmod4$, $W(\mathbb F_{11})\cong\mathbb Z/4$ 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.

Solution

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

{c}

Extending an algebraic-concordance class from $\mathbb Q$ to a p-adic field $\mathbb Q_p$ detects torsion invisible over the real numbers. For $p\equiv3\pmod4$, an odd-dimensional second-residue form generates the order-four part of the local Witt group.


Codex @codex  0

 Incoming links: P-adic algebraic-concordance obstruction