Kirby Problem 1.42

Definitions and mathematical statement

A Seifert form is an integral bilinear form $V$ on a free abelian group of even rank for which $V-V^T$ is unimodular. On a Seifert surface $F$ of a knot it is given by $V(a,b)=\operatorname{lk}(a^+,b)$ for homology classes represented by curves, where $a^+$ is a positive normal push-off. A form on a rank-$2g$ group is metabolic if a rank-$g$ direct summand $M$ has $V|_{M\times M}=0$. Forms $V_1,V_2$ are algebraically concordant if $V_1\oplus(-V_2)$ is metabolic.

Do algebraically concordant forms exist such that no pair of smoothly concordant knots $K_1,K_2$ realizes them as Seifert forms? Ask the same question for topological locally flat concordance. Concordance means an embedded annulus in $S^3\times[0,1]$ joining the knots, with the relevant smoothness condition.

The quantifier is over all realizations of the two forms: exhibiting one nonconcordant pair of knots is insufficient. The problem asks whether prescribed algebraic data can itself obstruct every concordant realization, even though the direct-sum form satisfies the usual algebraic concordance condition.

Short English statement

Can two algebraically concordant Seifert forms nevertheless be impossible to realize by concordant knots?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2701 (KP-1.42), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.42. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.