Problem 4.18 — Define a skein module of 3-manifolds, and calculate it.

Definitions and mathematical statement

Define an appropriate skein-module theory for the three-manifolds intended by the source and compute the resulting module. The extract supplies no prescribed objects, coefficient ring, local relation, or calculation target.

For context, a skein construction fixes a coefficient ring $R$, a class $\mathcal O$ of topological objects, and relations $\mathcal R$ expressed by linear combinations of local replacements, giving \[ \mathcal S=R\mathcal O/\langle\mathcal R\rangle. \] One reading uses embedded links/graphs in a fixed three-manifold $M$ to define $\mathcal S(M)$. Another uses three-manifolds themselves as generators related by specified cut-and-paste/surgery operations. The phrase "of 3-manifolds" does not distinguish these readings in the supplied item.

A complete proposal must identify its topological category, generators, local moves, coefficients, naturality/invariance requirements, and the sense in which it is to be computed. It must explain the intended nontrivial content rather than satisfy the words by an arbitrary zero or unrestricted free module. No particular existing link skein module or three-manifold surgery quotient is silently substituted for this definition-building request.

Short English statement

Define and calculate the intended three-manifold skein theory. The extract does not identify whether links in manifolds or manifolds themselves are the generators, or specify relations and useful invariance requirements.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 4.18, PDF page 82, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; accessed 2026-07-29, PDF-font-structure extraction.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400096$, source code catalogue entry, Problem 4.18. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.