Virtual-knot problem 27 — Study concordance and cobordism invariants of virtual knots.

Definitions and mathematical statement

Let $\mathcal V$ be virtual knots modulo the intended virtual Reidemeister and detour moves. In the usual oriented movie description, virtual cobordisms also allow births, deaths and saddle moves, and concordance is represented by a connected genus-zero cobordism between the two knots. The exact allowed surface/stabilization and framing conventions must be fixed.

The source asks for concordance and cobordism invariants of virtual knots, and specifically a classification of \[ \mathcal V/\!\sim_{\mathrm{pass}}, \] where $\sim_{\mathrm{pass}}$ is generated by virtual equivalence and the direct virtual generalization of the classical pass move intended by the source. A complete classification should give explicit realizable invariant values that are equal precisely on pass-equivalent knots.

The motivating classical comparison is pass equivalence classified by the Arf invariant. For a classical knot, this is the ${\mathbb{Z}}/2{\mathbb{Z}}$-valued Arf invariant of the quadratic form obtained from a Seifert form modulo $2$. The problem does not assert that the virtual quotient has only two classes or that this classical construction automatically extends to all virtual knots. No pass-move diagram is supplied, so the precise virtual relation cannot be recovered solely from its name.

Short English statement

Develop virtual-knot concordance and cobordism invariants, especially a complete classification under the intended virtual pass moves, using the classical Arf classification as motivation.

Sources

• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 27. https://arxiv.org/abs/1409.2823. Recorded access 2026-07-29; extraction source-tex. No individual proposer, pass-move figure or separate reference is supplied.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10600027$, exact source code catalogue entry. Its partially_solved/PARTIAL-PROGRESS summary mentions 2016 and 2024 studies of pass/Arf-type invariants and active concordance work, without complete citations or a documented full classification.