Virtual-knot problem 9 — Virtual braids

Definitions and mathematical statement

Let $VB_n$ be the virtual braid group on $n$ strands. In the usual birack coloring action, a birack has an invertible switch $S:X^2\to X^2$ satisfying the set-theoretic Yang--Baxter equation and the required birack nondegeneracy axioms. Classical generators act by $S$ on adjacent coordinates, inverse crossings by $S^{-1}$, and virtual crossings by the corresponding virtual switch, ordinarily the coordinate flip. This gives an action \[ \rho_n:VB_n\longrightarrow\operatorname{Bij}(X^n) \] when the defining mixed relations are satisfied. The exact source birack/virtual-switch convention is not included.

The first question is whether there exists a birack whose virtual-braid action is faithful: $\ker\rho_n=\{1\}$, so distinct braids induce distinct maps. The excerpt does not say whether a single birack must work simultaneously for all strand numbers or a separate choice is allowed for each fixed $n$. No finiteness of $X$ is required.

The second question is whether a particular virtual-braid invariant cited in the original survey is faithful. Both citations in that question were lost in extraction, so the invariant cannot be identified or replaced by a different representation.

The source states that the action defined by linear biquandles is not faithful, and says this "almost certainly" means the corresponding linear invariants of virtual knots and links are not faithful. The latter is retained as a qualified expectation, not a proved consequence: faithfulness of a braid action and completeness of a closed-knot/link invariant are distinct properties. No unnamed braid invariant is declared equal to a linear-biquandle invariant.

Short English statement

Does some birack distinguish all virtual braids through its action, and is the separately cited braid invariant faithful? The latter citations are missing. The source's expectation about closed virtual-knot invariants remains explicitly conjectural.

Sources

• Local UnsolvedMath registry, permanent ID 10600009, source code catalogue entry; no proposer or proposal year supplied; updated 2026-08-06.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey problem-list item 9: https://arxiv.org/abs/1409.2823. Registry access date 2026-07-29; source-TeX extraction. Provenance reports 60 top-level items with compound items retained. Present primary-literature status and redistribution terms require review.
• The original statement has empty citation slots for its second invariant question and its closing comparison; no bibliography keys are supplied.
• The summary lists "Representations of virtual braids by automorphisms and virtual knot groups" (Bardakov et al., 2016, arXiv:1603.xxxx); "Virtual and universal braid groups, their quotients and representations" (2021, arXiv:2107.xxxx); "Representations of flat virtual braids by automorphisms of free group" (2023, arXiv:2306.xxxx); "Maps from braids to virtual braids and braid representations" (2022, arXiv:2210.xxxx); and "The nontrivial kernel of Manturov-Nikonov map from classical braids to virtual braids" (2026, arXiv:2603.xxxx). All five identifiers contain placeholders and are not valid verifiable arXiv references as supplied; none is presented as a checked citation.