Virtual-knot problem 54 — Can one construct a projection from the set of graph-links to the set of realizable graph-links?

Definitions and mathematical statement

Let $\mathcal G$ be the set of graph-link equivalence classes in the source's graph-Reidemeister theory, and let $\mathcal R\subseteq\mathcal G$ be the realizable classes. In the customary chord-diagram interpretation, a realizable representative is an appropriately labelled graph obtained as an intersection graph of a chord diagram: vertices correspond to chords and adjacency means that their endpoints alternate. The excerpt omits the labels, graph moves and exact realizability convention; in particular it must be stated whether membership of $\mathcal R$ means that a class has at least one realizable representative.

The source asks to construct a projection onto realizable graph-links. The standard retraction reading is a map \[P:\mathcal G\to\mathcal R,\qquad P(r)=r\quad(r\in\mathcal R).\] With inclusion $i:\mathcal R\hookrightarrow\mathcal G$, this says $P\circ i=\operatorname{id}_{\mathcal R}$, and $i\circ P$ is idempotent. A rule on graph representatives must respect all graph-link equivalences to define such a map on classes.

The intended additional meaning of "construct" or "projection" is missing. A nonempty subset of a set admits a set-theoretic retraction with no useful geometric content, so the substantive problem needs, for example, an effective diagram rule, compatibility with graph operations, a prescribed relation to the original graph, or some other stated property. None of these examples is imposed as a requirement not present in the source. Existence of a set map, decidability of realizability of classes, and a local/natural projection are different tasks; the excerpt does not choose among them.

Short English statement

Construct a well-defined projection from graph-link classes onto the realizable ones, fixing realizable classes. The problem needs a further constructive or structural condition to distinguish its intended projection from a bare set retraction.

Sources

• Local UnsolvedMath registry, permanent ID $10600054$, exact source code catalogue entry.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 54: https://arxiv.org/abs/1409.2823. Registry access 2026-07-29; source-tex extraction. No separate primary construction reference is supplied.