Virtual-knot problem 57 — Is it true that two equivalent realizable graph-links are equivalent in the class of realizable graph-links?

Definitions and mathematical statement

Let $\mathcal G$ be the graph-link diagrams with the source's labels and graph Reidemeister moves, and let $\mathcal R\subseteq\mathcal G$ be the realizable diagrams. In the usual graph-link interpretation, realizability means representability by the appropriate chord diagram: graph vertices correspond to chords and adjacency to linked, alternating endpoints, with the required labels also respected.

Write $G\sim H$ if a finite sequence of all allowed graph-link moves joins $G$ to $H$, and $G\sim_{\mathcal R}H$ if such a sequence can be chosen with every intermediate diagram in $\mathcal R$. The question is whether \[ \forall G,H\in\mathcal R,\qquad G\sim H\ \Longrightarrow\ G\sim_{\mathcal R}H. \] Equivalently, does inclusion induce an injection $\mathcal R/\!\sim_{\mathcal R}\ \to\mathcal G/\!\sim$? If not, construct realizable endpoints joined by an unrestricted move sequence but by no sequence staying realizable. The requirement is existence of an entirely realizable path, not that every unrestricted path be realizable. The precise labelled graph and move conventions are not supplied by the excerpt.

Short English statement

Can any two equivalent realizable graph-link diagrams be connected through realizable diagrams alone? If not, exhibit a pair for which every connecting sequence must leave that class.

Sources

• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 57. https://arxiv.org/abs/1409.2823. Recorded access 2026-07-29; extraction source-tex. No individual proposer, graph-move definition or additional reference is supplied.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10600057$, exact source code catalogue entry; supplied OPEN-TRIAGE status and summary reporting no resolving literature found.