Problem 4.2 — (J.

Definitions and mathematical statement

In the standard Kauffman-bracket convention, let $R={\mathbb{Z}}[A,A^{-1}]$ and let $S_{2,\infty}(M)$ be the $R$-module generated by isotopy classes of unoriented framed links in a three-manifold $M$, including the empty link, modulo the local skein relation \[ [L_\times]=A[L_0]+A^{-1}[L_\infty] \] and the trivial-circle relation $[L\sqcup\bigcirc]=-(A^2+A^{-2})[L]$, with compatible local pictures/framing conventions. An element $x\ne0$ is torsion if $f(A)x=0$ for some nonzero $f(A)\in R$.

The source, attributed to J. Przytycki, reports that incompressible tori and two-spheres in $M$ yield torsion and asks whether other surfaces can yield torsion as well. The exact hypotheses and construction behind that reported fact are deferred to reference [339], which is absent. In particular essentiality and separation/embedding conditions cannot be inferred for arbitrary spheres or tori from the short sentence.

The requested extension concerns a geometric source of torsion, not merely a manifold that both contains another surface and has torsion from an unrelated torus or sphere. The source does not specify which additional surface types, their incompressibility or proper/closed status, or what construction constitutes yielding torsion. A precise positive instance would identify the surface-induced relation, its nonzero annihilating polynomial and the nonvanishing of the module class. No universal assertion that every higher-genus surface creates torsion is supplied.

The coefficient ring and manifold category are not reproduced. The displayed Laurent-polynomial skein convention is explicit standard context; specialization or changing coefficients can change the torsion question and is not silently allowed.

Short English statement

Can surfaces beyond the source's torus and sphere cases geometrically produce torsion in the Kauffman-bracket skein module? Specify the surface conditions and exhibit a nonzero annihilated class, rather than only coexisting torsion.

Sources

• Local UnsolvedMath registry, permanent ID 10400080, exact source code catalogue entry, Problem 4.2; source attribution J. Przytycki, no proposer/year in metadata. Registry status open, classification OPEN-TRIAGE; title truncated.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 4.2, PDF page 74, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied pdf-font-structure extraction accessed July 29, 2026; provenance distinguishes 231 bold numbered items. No PDF was accessed here.
• The source cites [339] for the torus/sphere torsion statement, but its bibliography entry and precise theorem hypotheses are absent. The supplied summary gives no later construction or resolution.
• Provenance flags current primary-literature status and redistribution terms NEEDS_REVIEW.