Kirby Problem 1.6

Definitions and mathematical statement

A Seifert matrix is an integral square matrix $V$ with $V-V^T$ unimodular. For a Seifert surface $F$ and a basis $a_i$ of $H_1(F;\mathbb Z)$, it is realized by $V_{ij}=\operatorname{lk}(a_i^+,a_j)$, where $a_i^+$ is pushed in the positive normal direction to $F$. Two such matrices are $S$-equivalent if related by integral unimodular congruences and elementary enlargements or their inverses. An enlargement adjoins two basis vectors $e,f$, with $f$ having zero pairings with the old module and with itself, and $(V(e,f),V(f,e))=(1,0)$ or $(0,1)$; the other new pairings are arbitrary.

Given any two $S$-equivalent matrices $V_1,V_2$, must there be one knot $K$ with Seifert surfaces $F_1,F_2$ realizing these matrices in suitable bases? The surfaces are not required to be disjoint or to have equal genus. The source also asks the stronger uniform realization question: for each entire $S$-equivalence class, does some single knot realize every form in that class? That assertion has a different quantifier order from the specified-pair question; both formulations keep the boundary knot fixed while the surface varies.

Short English statement

Can two algebraically equivalent Seifert forms be realized by surfaces bounding the same knot?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2665 (KP-1.6), 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 1.6. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.