Virtual-knot problem 50 — The problem about cobordisms in sections: Let us have a free knot and its cobordism.

Definitions and mathematical statement

A free-knot diagram is conventionally a framed $4$-valent graph with one unicursal component, where the framing specifies opposite half-edges at each vertex; diagrams are taken modulo free Reidemeister moves. Let $D$ represent the given free knot and let $W$ be its given cobordism, in the source's unspecified cobordism category.

A usual ${\mathbb{Z}}/2{\mathbb{Z}}$-valued parity assigns each vertex $v$ a label $p(v)\in{\mathbb{Z}}/2{\mathbb{Z}}$, compatibly with moves: a vertex removed by a first move has label zero, the two vertices of a second move have equal labels, and corresponding labels in a third move agree with the sum of the three involved labels zero. This explains a standard parity notion; the source does not state its coefficient group or axioms.

Problem-list item 50 asks to construct a parity on the given free knot using only the given cobordism and respecting only moves inside that cobordism. Schematically, it seeks labels $p_W(v)$ intrinsic to the specified cobordism data and compatible with the induced, permitted changes of sections/diagrams in $W$. It does not require compatibility with every unrelated cobordism or every unrestricted boundary move. The permitted interior moves, the section structure, and the equivalence on $W$ are not defined in this extract; even whether $W$ is a null-cobordism or has another boundary knot is unspecified.

The record gives no nontriviality or detection requirement. Since an identically zero assignment satisfies the usual parity axioms, a precise intended research problem needs additional source criteria distinguishing the sought cobordism-dependent construction. No such criterion or particular geometric rule is invented here.

Short English statement

Use a specified free-knot cobordism itself to define parity labels, compatible with the moves allowed inside it. The cobordism and section-move conventions are missing, as is a condition excluding a vacuous parity. The supplied assessment describes an ongoing parity-cobordism program without documenting this precise construction as completed.

Sources

• Local UnsolvedMath registry, permanent ID $10600050$, exact source code catalogue entry, virtual-knot problem-list item 50, "The problem about cobordisms in sections." No individual proposer or proposed year is supplied; the title is truncated.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, dedicated problem-list item 50: https://arxiv.org/abs/1409.2823. Background: 60 top-level items, compound items retained intact; source-TeX extraction, accessed 2026-07-29.
• Supplied incomplete references: "Parity and Cobordisms of Free Knots" (2010), arXiv:1001.xxxx; "Cobordisms of Free Knots and Gauss Words" (2009), arXiv:0904.xxxx; "A Sliceness Criterion for Odd Free Knots" (2017), arXiv:1707.xxxx. The xxxx identifiers are placeholders and have not been completed.
• Source evidence NEEDS_REVIEW presents an open problem and calls for a current primary-literature audit. Supplied OPEN-TRIAGE/borderline summary says the parity-cobordism program exists and may address the construction, but completion of this particular request is not documented as a closed problem.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.