Problem 12.23 — (A.

Definitions and mathematical statement

An oriented link $L\subset S^3$ is positive if it admits a diagram in which every crossing is positive. Let $\sigma(L)$ be its ordinary link signature, using one fixed sign convention. For an admissible Seifert surface $S$ with oriented boundary $L$, write $\chi(S)$ for its Euler characteristic, and define \[ \chi_{\max}(L)=\max\{\chi(S):S\text{ an admissible Seifert surface for }L\}. \] The usual definition excludes closed components. Whether surfaces must be connected, and how split links are handled, is not specified in the source and affects this convention.

A. Stoimenow's Problem 12.23 asks whether, for every fixed signature value $s$, there is a lower bound depending only on $s$: \[ \forall s\in{\mathbb{Z}}\ \exists b_s\in{\mathbb{Z}}\ \forall\text{ positive links }L:\quad \sigma(L)=s\ \Longrightarrow\ \chi_{\max}(L)\ge b_s. \] Thus the lower bound is on the maximal surface Euler characteristic, not an upper bound on it and not a lower bound for every spanning surface. No bound on the number of components or restriction to knots appears in the question. For knots, with the usual connected-surface convention, $\chi_{\max}(K)=1-2g(K)$, so this becomes an upper-genus-bound question at fixed signature; that special case is not substituted for the full link question.

Short English statement

At each fixed signature, do all positive links have their maximal Seifert-surface Euler characteristic bounded below by a constant depending only on that signature? The source's surface-connectivity and signature conventions need checking; the question includes links with unrestricted component count.

Sources

• Local UnsolvedMath registry, permanent ID $10400228$, exact source code catalogue entry, Problem 12.23, attributed to A. Stoimenow. The title is truncated after "(A."
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.23, PDF page 169: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold headings distinguish 231 numbered items from references in surrounding exposition.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and calls for a current primary-literature audit. Supplied classification OPEN-TRIAGE identifies the fixed-signature lower-bound question without a further reference or resolution.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.