Conjecture 3.22 — [357, 139] The map (29) is injective.

Definitions and mathematical statement

The conjecture asserts injectivity of the source's map (29), and in particular injectivity of its map (30). Neither displayed map, domain, codomain, nor surrounding definitions are supplied.

Writing only placeholders $F_{29}:A_{29}\to B_{29}$ and $F_{30}:A_{30}\to B_{30}$, the retained claims are \[ \forall u,v\in A_{29}:\quad F_{29}(u)=F_{29}(v)\Longrightarrow u=v, \] and the analogous implication for $F_{30}$. The source asserts the second as a consequence of the first, but the relation between the maps is absent and is not reconstructed.

The summary calls this an injectivity problem for a loop-expansion map. It does not identify the diagram algebra, coefficients, localizations, completions or formula, so no particular hair/loop map or knot-detection assertion is substituted. Internal references [357, 139] are not resolved in the supplied record. The summary's open/triage and "not obviously resolved" assessment is qualified source information, not a verified current theorem status.

Short English statement

Prove that the maps numbered (29) and consequently (30) in the source are injective. Their definitions and referenced bibliography are missing, so the exact loop-expansion claim cannot be recovered here.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400072$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Conjecture 3.22, PDF page 70, referring to maps (29), (30) and bibliography labels [357, 139]: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29.
• The actual equations and bibliographic entries behind [357, 139] are not included; no separate resolution reference is supplied.