Problem 5.8 — Compute H 2 Q(X; A) for each X -module A.

Definitions and mathematical statement

Let $X$ be the source's quandle. A quandle operation is idempotent, has bijective right translations, and satisfies right self-distributivity. For each $X$-module $A$ in the source's coefficient category, the task is to compute the second quandle cohomology group \[ H_Q^2(X;A). \] Once its coefficient cochain complex is specified, this means identifying the quotient of normalized 2-cocycles by 2-coboundaries, \[ H_Q^2(X;A)=\ker(\delta:C_Q^2(X;A)\to C_Q^3(X;A))\big/ \operatorname{im}(\delta:C_Q^1(X;A)\to C_Q^2(X;A)), \] not merely giving the tautological kernel/image notation or isolated cocycles.

The excerpt does not define $X$, the $X$-module structure on $A$, the coefficient action or differential, and the normalization relations. Coefficients may carry nontrivial quandle-dependent structure, so ordinary trivial abelian coefficients cannot be substituted. Quandle cohomology must likewise be distinguished from rack cohomology before imposing degeneracy relations.

The quantifier is every admissible $X$-module $A$, with no finiteness or field restriction supplied. A calculation only of rational dimensions or only for trivial modules would therefore address a narrower task. The surrounding source is needed to tell whether $X$ is fixed earlier or ranges over a specified class, and to define the actual cochain complex.

Short English statement

Compute degree-two quandle cohomology for every module over the intended quandle, retaining its coefficient action and torsion. The source excerpt omits the quandle/module definitions and the resulting differential.

Sources

• Local UnsolvedMath registry, permanent ID 10400104, source code catalogue entry; no proposer or proposal year supplied; update 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 5.8, PDF page 92: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access date 2026-07-29; PDF font-structure extraction. Provenance distinguishes 231 numbered bold headings. Current primary-literature status and redistribution terms require review.
• The supplied summary repeats the module-coefficient calculation question without precise additional references or verified computations. Adjacent local records do not restore the omitted module/cochain definitions.