Problem 4.4 — (J.

Definitions and mathematical statement

Let $F_{0,3}$ be a pair of pants, the compact orientable genus-zero surface with three boundary components, and set $M=F_{0,3}\times S^1$. Its three torus boundary components are retained.

In the usual Kauffman-bracket notation, $\mathcal S_{2,\infty}(M)$ is generated by isotopy classes of unoriented framed links in $M$, including the empty link, modulo \[ [L_\times]=A[L_0]+A^{-1}[L_\infty],\qquad [L\sqcup O]=-(A^2+A^{-2})[L], \] where the links in the first relation agree outside a ball and differ there by one crossing and its two standard smoothings, and $O$ is an isolated zero-framed trivial circle. Over a general ground ring, $A$ must be invertible; the universal common choice is $\mathbb Z[A^{\pm1}]$.

Compute \[ \mathcal S_{2,\infty}(F_{0,3}\times S^1) \] as a module over the intended coefficient ring, giving its structure rather than only a numerical rank unless the source asks for a field specialization. The extract does not state the coefficient ring, parameter specialization, or localization convention. In particular, replacing the universal module by its tensor product with $\mathbb Q(A)$ can discard torsion and is not silently done. The displayed skein relations explain the standard interpretation of the notation and should be matched to the source convention.

Short English statement

Compute the Kauffman-bracket skein module of pair-of-pants times a circle, retaining the boundary and specifying the ground ring so that torsion is not lost by an unstated field localization.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 4.4, PDF page 75, attributed to J. Przytycki, 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 $10400082$, source code catalogue entry, Problem 4.4. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.