Problem 4.16 — (J.

Definitions and mathematical statement

For a link diagram, a Fox 7-coloring assigns an element of $\mathbb F_7$ to every arc, with crossing condition \[ a+c=2b\quad\text{in }\mathbb F_7, \] where $b$ is the overpassing color and $a,c$ are the two underpassing colors. Let $N_7(L)$ be the number of such assignments in the usual link convention, counting constant colorings. The source does not specify reduced/nonconstant or color-symmetry quotients, so those are not silently substituted.

The problem asks which coefficient choices for the source's $(4,\infty)$ skein relation make the number of Fox 7-colorings "measured by" its skein module. The relation is referred to through Figure 16, but neither the figure, tangle terms, coefficients nor framing relation is supplied.

At minimum a factorization interpretation would require, for the chosen skein module $S_{4,\infty}$, a well-defined measurement $\Phi$ with \[ \Phi([L])=N_7(L)\qquad\text{for every admitted link }L. \] If $\Phi$ is required to be linear, its coefficient specialization and target module must also be fixed; a set-valued factorization alone is weaker. The source does not define "measured", so no linearity, rank or cardinality interpretation is selected. Determine all coefficient systems satisfying the intended version, after those conventions are recovered.

The ambient manifold, coefficient ring and framing/normalization are likewise absent. The summary only reports this as not obviously resolved, with no exact modern result. The 2002 question and later open/triage label remain qualified source attributions pending primary-literature review.

Short English statement

Which coefficients in the specified $(4,\infty)$ skein relation let its module recover the Fox 7-coloring count? The relation's figure and the meaning of measured are missing, so the exact coefficient classification cannot yet be stated.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400094$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 4.16, attributed to J. Przytycki, PDF page 80, referring to Figure 16: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items.
• Figure 16 and exact relation coefficients are not supplied, and no separate resolution reference is included.