Problem 5.6 — Compute the quandle cocycle invariant Φ α(K) of each knot K for a second cohomology class α of a connected quandle.

Definitions and mathematical statement

A quandle $(Q,*)$ satisfies $x*x=x$, each right translation $x\mapsto x*y$ is bijective, and $(x*y)*z=(x*z)*(y*z)$. It is connected if the group generated by right translations acts transitively. For an abelian coefficient group $A$, written multiplicatively, a normalized quandle 2-cocycle $\theta$ satisfies \[ \theta(x,x)=1,\qquad \theta(x,y)\theta(x*y,z)=\theta(x,z)\theta(x*z,y*z). \] Let $\alpha\in H_Q^2(Q;A)$ be its cohomology class.

In the standard finite-quandle convention, color an oriented knot diagram by $Q$ using the quandle crossing rule, and define \[ \Phi_\alpha(K)=\sum_{\mathcal C\text{ a }Q\text{-coloring}} \prod_{c\text{ a crossing}}\theta(a_c,b_c)^{\varepsilon_c} \quad\in\mathbb Z[A]. \] Here $b_c$ is the over-arc color, $a_c$ the source under-arc color in the chosen coorientation convention, and $\varepsilon_c$ the crossing sign. This requires a consistent convention at negative crossings; the excerpt does not reproduce it.

The source asks to compute these cocycle invariants for each knot and the intended second cohomology classes of connected quandles. It does not specify a particular quandle, coefficient group, knot family or output form. Finiteness is not stated: an infinite connected quandle may yield infinitely many colorings, so the displayed ordinary group-ring state sum needs an additional restriction or definition in that setting. For finite input data the defining sum is enumerable; the source's computation program must therefore be distinguished from a new claim of basic computability. No closed formula, efficiency target or complete table scope is supplied.

The summary reports many special-knot/quandle computations but gives no examples or citations. Its partial-progress label remains source information pending current primary-literature review.

Short English statement

Compute the quandle 2-cocycle state-sum invariant for the intended knots, connected quandles and cohomology classes. The coefficient group, finite/infinite convention and desired computational output are missing, so no universal formula or new computability claim is inferred.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400102$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 5.6, PDF page 89: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items.
• No separate special-case computation or current resolution reference is supplied.