Problem 4.11 — Calculate the higher skein modules based on the Kauffman skein relation W 3,∞ i (M ) and ˆW 3,∞(M ) (see below for the…

Definitions and mathematical statement

Problem 4.11 asks to calculate the higher skein modules based on the Kauffman skein relation, denoted in the source by \[ \mathcal W_i^{3,\infty}(M)\qquad\text{and}\qquad \widehat{\mathcal W}^{3,\infty}(M). \] Both named constructions are required. The excerpt explicitly says their definitions occur below, but that defining passage is not supplied. Thus the coefficient ring, generators, relations or chain constructions, index range, meaning of "higher", and significance of the hat are unknown. In particular the hat is not automatically a degree completion and the index is not automatically homological degree.

The target is a module calculation in the source's conventions for the intended 3-manifolds $M$, including any torsion rather than only dimensions after scalar extension. The class of manifolds and the acceptable form of a calculation are also unspecified. These $(3,\infty)$ higher Kauffman constructions must not be silently replaced by the ordinary $(2,\infty)$ Kauffman-bracket skein module, a selected skein homology, or Khovanov homology. A self-contained formulation needs the missing definitions before any exact module isomorphism can be stated.

Short English statement

Compute both higher Kauffman skein-module constructions named in the source. Their definitions, coefficient ring, index and hat conventions, and manifold scope are absent from the extract.

Sources

• Local UnsolvedMath registry, permanent ID 10400089, source code catalogue entry; no proposer or proposal year supplied; updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 4.11, PDF page 77: 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. The definitions referenced as "below" are not included. Present status and redistribution terms require review.
• The supplied summary repeats the calculation task without a precise additional reference or verified computation.