Problem 4.13 — Calculate HS q(M ) for each 3-manifold M.

Definitions and mathematical statement

Problem 4.13 asks: \[ \text{For every }3\text{-manifold }M,\quad \text{calculate }\mathrm{HS}_q(M). \] The notation $\mathrm{HS}_q$ and the parameter $q$ have no definition in the supplied extract. In particular, no coefficient ring, link equivalence, local skein relations, or homotopy relations are given. The typography suggests a parameterized skein construction, but it does not establish which one, so no particular homotopy skein module is silently substituted.

If the intended object is a module defined by generators and relations, a meaningful calculation would determine it up to module isomorphism over the source's specified coefficient ring, or give an explicit presentation with that ring and all relations fixed. That explains a possible form of output without defining the missing invariant. The universal quantifier in the record is over $3$-manifolds with no stated orientability, compactness or boundary restriction. These conventions, the status of $q$ as a formal variable or specialized scalar, and the intended notion of calculation must be recovered from the source before a precise mathematical target is available.

Short English statement

Compute the source's object $\mathrm{HS}_q(M)$ for every three-manifold. Its definition and parameter conventions are absent, so the draft records the requested universal calculation without assigning a different skein theory to the symbol.

Sources

• Local UnsolvedMath registry, permanent ID $10400091$, exact source code catalogue entry, Problem 4.13. No individual proposer or proposed year is supplied.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 4.13, PDF page 78: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold printed headings distinguish 231 numbered items from surrounding references.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied OPEN-TRIAGE summary calls the all-manifolds computation not obviously resolved. No further reference defining $\mathrm{HS}_q$ is supplied.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.