Virtual-knot problem 52 — If $X$ is the free rack, is $\Gamma X$ a cat(0) space?

Definitions and mathematical statement

A rack is a set with bijective right translations and right self-distributive operation. The free rack $F_{\mathrm{Rack}}(S)$ on a set $S$ has the property that every map from $S$ to a rack extends uniquely to a rack homomorphism. Let $X$ be the free rack intended by the source, and retain its notation $\Gamma X$ for the associated space.

The question is whether $\Gamma X$, with the intended metric, is CAT$(0)$. A CAT$(0)$ space is a geodesic metric space in which distances between points on sides of any geodesic triangle are at most the corresponding distances in its Euclidean comparison triangle. The source states that a positive answer would imply \[ \pi_n(\Gamma X,x)=0\qquad(n\geq2) \] for every relevant basepoint $x$.

The construction $\Gamma$, the number or set of free generators, and the metric are absent from the excerpt. It is therefore unresolved whether the question concerns a specified cubical metric, existence of some metric, a local nonpositive-curvature condition, or a universal cover. These are distinct targets; none is substituted for the printed global CAT$(0)$ question.

Short English statement

Is the source's space $\Gamma X$ associated to a free rack CAT$(0)$? The intended construction and metric must be supplied; the reported consequence is vanishing higher homotopy.

Sources

• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 52. https://arxiv.org/abs/1409.2823. Recorded access 2026-07-29; extraction source-tex. No definition reference for $\Gamma$, individual proposer or further citation is supplied.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10600052$, exact source code catalogue entry; supplied OPEN-TRIAGE status and summary reporting no literature found.