Problem 12.21 — (Y.

Definitions and mathematical statement

Let $K\subset S^3$ be a knot and $V$ a Seifert matrix. Its determinant and classical signature are, with a fixed sign convention, \[ \det(K)=|\Delta_K(-1)|=|\det(V+V^{\mathsf T})|,\qquad \sigma(K)=\operatorname{sign}(V+V^{\mathsf T}), \] where $\Delta_K$ is the Alexander polynomial and the sign of a real symmetric matrix is its number of positive eigenvalues minus its number of negative eigenvalues. The determinant is unaffected by the usual unit ambiguity in $\Delta_K$.

Y. Shinohara asks whether \[ \forall k\in{\mathbb{Z}}_{>0}\quad\exists K\subset S^3:\qquad \det(K)=4k+1\quad\text{and}\quad\sigma(K)=4. \] The knot may depend on $k$. No prime, alternating, genus or other subclass restriction is given. If the source uses the opposite signature sign convention, mirroring interchanges signatures while preserving determinant, so the stated existence question is unchanged.

Short English statement

For every positive integer $k$, is there a knot with determinant $4k+1$ and classical signature four? No additional knot class is imposed.

Sources

• Local UnsolvedMath registry, permanent ID 10400226, source code catalogue entry; statement attribution Y. Shinohara, source reference [364]; proposer field and proposal year absent; updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.21, PDF page 168: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access date 2026-07-29; PDF font-structure extraction. Provenance distinguishes 231 numbered bold headings. Present primary-literature status and redistribution terms require review.
• Reference [364] is not supplied with its bibliography entry. The research summary repeats the question without a precise later reference or verified resolution.