Virtual-knot problem 24 — Does it follow that $K$ and $K'$ are equivalent as classical knots?

Definitions and mathematical statement

Let $\mathcal K$ be the set of classical knot types under classical Reidemeister equivalence, and let $\mathcal V$ be virtual knot types under the usual classical and virtual Reidemeister/detour equivalences. Write $\sim_Z$ for the further equivalence generated by the source's $Z$-moves, and write $\mathcal V/\!\sim_Z$ for the quotient denoted $VKT/Z$ in the source. The actual local $Z$-move is referred to an earlier section and is not included in the supplied record; this notation is consequently conditional on that source definition.

Problem-list item 24 asks whether the natural map \[ \mathcal K\longrightarrow\mathcal V/\!\sim_Z \] is injective. Equivalently, for every pair of classical knot diagrams $K,K'$, does equivalence by virtual knot moves together with $Z$-moves imply that $K$ and $K'$ are equivalent as classical knots?

The source states that classical knot theory embeds in ordinary virtual knot theory, and that $Z$-equivalent diagrams have equal Jones polynomials. These are supplied background assertions; equality of Jones polynomials alone is not the proposed injectivity statement. Orientation and mirror conventions should be inherited from the omitted source definitions, rather than added as extra quotient relations.

Short English statement

If two classical knots become equivalent after allowing virtual moves and the source's $Z$-moves, must they already be classically equivalent? The local definition of a $Z$-move is missing from this excerpt.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10600024$, exact source code catalogue entry.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 24: https://arxiv.org/abs/1409.2823. Supplied access date: 2026-07-29; extraction: source-tex. Its 60 top-level items retain compound questions intact.
• The statement refers to an earlier section for $Z$-equivalence; that section and its move diagrams are not part of this record. No additional primary reference supports the registry's unsuccessful literature-search assessment.