Virtual-knot problem 38 — The Kauffman bracket polynomial of a virtual diagram can have the leading term, i.e., the term having the highest pos…

Definitions and mathematical statement

A virtual link diagram has classical crossings, carrying over/under data, and virtual crossings. Its virtual isotopy class is generated by the generalized Reidemeister moves, including the virtual detour moves. The source asks for a move description within a restricted class of such diagrams; it does not specify whether only one-component diagrams are intended.

The phrase "leading term equal to zero" refers to cancellation at an a priori highest possible degree, rather than to the actual highest nonzero coefficient of a polynomial. For example, in the usual normalization for a nonempty diagram, \[ \langle D\rangle=\sum_s A^{a(s)-b(s)}(-A^2-A^{-2})^{|s|-1}, \] where the states smooth the classical crossings, $a(s)$ and $b(s)$ count the two smoothing types, and $|s|$ counts the resulting state circles. If $c(D)$ is the number of classical crossings and $s_A$ is the all-$A$ state, the customary candidate upper degree is \[ M(D)=c(D)+2(|s_A|-1). \] Under this convention, write \[ \mathcal C=\{D:[A^{M(D)}]\langle D\rangle\ne0\}. \] The source does not explicitly give this normalization or degree formula; the intended highest-possible-degree convention must be checked against the survey before treating this displayed definition as definitive. In particular, the condition is stronger than merely requiring a nonzero bracket polynomial.

Problem. Classify moves, each a composition of generalized Reidemeister moves, that establish virtual isotopy equivalence within the class of diagrams having nonzero highest-possible-degree term. A complete move description should connect any two diagrams $D,D'\in\mathcal C$ representing the same virtual isotopy class by a sequence of the specified moves with diagrams in $\mathcal C$ between the specified moves. The source does not state whether every elementary intermediate diagram inside a composite move must also belong to $\mathcal C$, nor whether a finite list of composite move types is required. These distinctions are unresolved parts of the formulation, rather than additional assumptions.

Short English statement

Describe the combinations of virtual Reidemeister moves needed to relate equivalent diagrams whose Kauffman brackets retain their expected top-degree term. The expected degree convention and the admissibility of intermediate diagrams need clarification.

Sources

• Local UnsolvedMath registry, permanent ID 10600038, source code catalogue entry, including its complete statement, provenance, and research summary.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 38; https://arxiv.org/abs/1409.2823. The registry records source-TeX extraction and access on 2026-07-29. No individual proposer or proposal year is supplied.
• The registry summary mentions "Local transformations and functorial maps" (2023) with the literal identifier arXiv:2301.xxxx, and unspecified parity-based bracket studies. The identifier is a placeholder, not a usable citation; the studies are not individually identified.