Virtual-knot problem 59 — Construct “graph-braids”.

Definitions and mathematical statement

The source asks to construct objects called "graph-braids." This term is not defined in the supplied item. In the surrounding graph-link/virtual-knot programme, a mathematical construction must specify what data a graph-braid carries, its equivalence relations, and what makes it a braid-type object; if there are strand numbers, composition or a closure operation, those must be defined and their relation to the relevant link theory stated. The excerpt does not itself prescribe such operations, a group rather than a monoid/category, a presentation, or an Alexander/Markov-type theorem. These cannot be supplied as unstated requirements.

The research summary expressly distinguishes this request from ordinary graph braid groups. For clarity, the latter concern configurations of distinct points on a fixed graph $\Gamma$, for example \[\operatorname{UConf}_n(\Gamma)=\{(x_1,\ldots,x_n)\in\Gamma^n:x_i\ne x_j\text{ for }i\ne j\}/S_n,\] and their fundamental groups. Constructing or citing those groups would not identify the "graph-braids" requested here. The actual graph-theoretic input and compatibility with graph-links or virtual links require the missing surrounding definitions. Thus this is an underspecified theory-construction request rather than a proposition about an already given class.

Short English statement

Define and construct the intended graph-braid objects in the survey's knot-theoretic setting. The record explicitly says that configuration-space braid groups on graphs are a different subject.

Sources

• Local UnsolvedMath registry, permanent ID $10600059$, exact source code catalogue entry.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 59: https://arxiv.org/abs/1409.2823. Registry access 2026-07-29; source-tex extraction. No separate primary definition/reference is supplied.
• The summary mentions unrelated configuration-space literature using only the placeholders "arXiv:0907.xxxx" (2009) and "arXiv:2210.xxxx" (2022); these are incomplete identifiers, not citations supporting the intended construction.