Problem 5.9 — Let the notation be as above.

Definitions and mathematical statement

Let $Q(K)$ be the fundamental quandle of a knot $K$; a coloring by a quandle $X$ is a quandle homomorphism $\rho:Q(K)\to X$. A lifting question requires a specified quandle map $p:Y\to X$. Given it, the lift set of $\rho$ is \[ \operatorname{Lift}_p(\rho)=\{\widetilde\rho:Q(K)\to Y\text{ a quandle homomorphism}:p\circ\widetilde\rho=\rho\}. \] Thus liftability is nonemptiness of this set, and in a finite setting its cardinality gives a possible refinement.

Using the missing preceding notation, extend the quandle cocycle invariant to a knot invariant associated with $\alpha$ that counts representations $Q(K)\to X$ while retaining information about whether each one lifts to $Q(K)\to Y$. The definition should recover the intended cocycle data, not just forget it and count ordinary colorings.

The source does not specify $\alpha$, its degree/coefficient object, the map or extension $Y\to X$, or how $\alpha$ governs the lifting obstruction. It also does not state finiteness hypotheses or the target of the enhanced invariant. Therefore neither a particular cocycle-extension formula nor a finite state sum is supplied as the answer. The following "Quantum quandles" section heading is separate material, not a quantum constraint on this problem.

Short English statement

Extend the $\alpha$-associated quandle cocycle invariant so that each knot-quandle coloring records whether it lifts through the intended $Y\to X$. The cocycle/extension data and output convention are missing.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 5.9, PDF page 92, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; accessed 2026-07-29, PDF-font-structure extraction. The problem explicitly refers to preceding notation absent from this record.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400105$, source code catalogue entry, Problem 5.9. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.