Problem 5.10 — (M.

Definitions and mathematical statement

A classical quandle is a set $X$ with a binary operation $*:X\times X\to X$ such that, for all $x,y,z\in X$, \[x*x=x,\qquad R_y:x\mapsto x*y\text{ is bijective},\qquad (x*y)*z=(x*z)*(y*z).\] The problem attributed to M. Polyak is to define a quantum quandle. This is a request to propose and justify an appropriate mathematical notion, not a conjecture about a structure already defined in the supplied record. A complete answer would have to state the underlying objects, operations and axioms and explain their relation to classical quandles. The source does not specify whether the desired relation is a deformation with a classical specialization, a noncommutative analogue, a categorical construction, or another quantization procedure. It supplies no coefficient ring, deformation parameter, classical-limit condition, morphisms, examples, or invariant-building requirement. Consequently the classical axioms above fix the object to be generalized but do not prescribe quantum axioms or assert that a particular existing meaning of the phrase meets the source's intention.

Short English statement

Polyak asks for a suitable definition of a quantum analogue of a quandle. The record does not specify what form of quantization or classical recovery the new definition should satisfy.

Sources

• Local UnsolvedMath registry, permanent ID $10400106$, exact source code catalogue entry.
• M. Polyak, attribution in Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 5.10, PDF page 92: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access 2026-07-29, extraction pdf-font-structure.