Virtual-knot problem 42 — One can consider braids with even numbers of strands.

Definitions and mathematical statement

Let $B_n$ denote the relevant group of $n$-strand braids and let $\widehat\beta$ denote the closure of $\beta$. The source does not state whether the intended category is classical, virtual, or another braid/link category; this choice must be fixed. Put \[ \mathcal B_{\mathrm{even}}=\coprod_{m\geq1}B_{2m}. \] In the usual Markov framework conjugation preserves strand number, while stabilization or destabilization changes it by one and hence changes its parity.

Question. Can Markov's theorem be reformulated using only braids with an even number of strands? A precise reformulation should give allowed moves on $\mathcal B_{\mathrm{even}}$ such that, for even-strand braids $\beta,\gamma$, \[ \widehat\beta\cong\widehat\gamma \quad\Longleftrightarrow\quad \beta\sim_{\mathrm{even}}\gamma, \] where $\sim_{\mathrm{even}}$ is generated by those moves and every braid occurring in a move sequence has even strand number. For a formulation covering the full intended link theory, every relevant closure class should also have an even-strand representative. The braid operations, closure equivalence, and scope of this coverage must be specified in the chosen category. The excerpt provides no proposed even-strand move set. It is not enough merely to require even endpoints in an ordinary sequence that still passes through odd-strand braids.

Short English statement

Find a Markov theorem whose representatives and all intermediate braids have even strand number, with moves detecting exactly equality of closures. The source does not select the braid category or propose the replacement moves.

Sources

• Local UnsolvedMath registry, permanent ID 10600042, source code catalogue entry, virtual-knot problem 42. No individual proposer is supplied.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 42. Supplied URL: https://arxiv.org/abs/1409.2823; recorded access 2026-07-29; extraction source-tex. The dedicated section contains 60 top-level enumerated items with compound items retained intact.
• The supplied summary lists "Virtual Braids" (2004), arXiv:0407.xxxx; "Virtual Braids and the L-Move" (2005), arXiv:0507.xxxx; "Markov Theorem For Free Links" (2011), arXiv:1112.xxxx; and "A Markov's theorem for extended welded braids and links" (2017), arXiv:1705.xxxx. These incomplete identifiers are retained as supplied, not treated as verified citations for the even-strand question.
• Supplied status open and OPEN-TRIAGE. The summary says no specific reformulation or even-strand treatment was located, while related Markov theorems exist. This is an editorial search assessment. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.