Virtual-knot problem 28 — Biquandles

Definitions and mathematical statement

Fix a convention for biquandles, namely the algebraic crossing operations and invertibility and Reidemeister axioms used by the source. The supplied excerpt does not list these axioms. In that category, a free biquandle on a set $X$ is a biquandle $F(X)$ with a map $\eta:X\to F(X)$ such that, for every biquandle $B$ and every map $f:X\to B$, there exists a unique biquandle homomorphism $\overline f:F(X)\to B$ with $\overline f\eta=f$.

Problem-list item 28 retains three questions:

1. Give a descriptive representation of the free biquandle, in addition to its abstract universal property or formal generators-and-relations description.
2. Give a topological explanation of the fundamental biquandle of a knot. In the usual diagrammatic construction, generators correspond to semiarcs and relations encode the crossing operations; the request is for a topological understanding of this object, with conventions compatible with that presentation.
3. Determine whether there is a useful notion of a free partial biquandle. The source expresses the tentative expectation "probably not," rather than a proved nonexistence assertion.

The meanings of partial biquandle and descriptive representation, the generator scope, and the desired intrinsic topological construction are not specified. Thus the last question has no fixed universal property until a category of partial objects and its morphisms is defined. The excerpt also leaves the intended classical or virtual knot scope and orientation conventions unstated.

Short English statement

Find a concrete description of free biquandles and a topological interpretation of knot biquandles, and investigate whether an appropriate free partial biquandle can exist. The source's negative expectation for the last question is tentative.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10600028$, exact source code catalogue entry, "Virtual-knot problem 28 --- Biquandles."
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 28: https://arxiv.org/abs/1409.2823. Supplied access date: 2026-07-29; extraction: source-tex. All three parts of this compound item are retained.
• The registry mentions general biquandle constructions with the incomplete identifier arXiv:1908.xxxx (2019). This is not a usable bibliographic citation; no precise additional reference or individual attribution is supplied.