RemarkRemark: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Jones polynomial unknot-detection status

Historical question

Assume full AC for the library Jones invariant and ${A C}_{ω}$ for the crossing number. A classical knot is a smooth embedding $S^{1} ↪ S^{3}$ considered up to ambient isotopy; $U$ is the unknot represented by a crossing-free circle, and “nontrivial” means it is not ambient isotopic to $U$ . A regular diagram is a generic plane projection with only transverse double points, each marked with over/under information. Let $c (K)$ be the least crossing count among regular diagrams representing $K$ . Let $V_{K} (x)$ be the library-normalized Jones polynomial with $V_{U} (x) = 1$ , where $x = t^{1 / 2}$ and $t = A^{- 4}$ . Set $p = x - x^{- 1}$ and write the unique expansion $V_{K} (x) = \sum_{q \geq 0} (a_{q} (K) + b_{q} (K) x) p^{q},$ where the integer coefficients have finite support; set $f_{q} (K) = (a_{q} (K), b_{q} (K))^{T}$ . The historical Jones unknot-detection question is whether $V_{K} (x) = 1$ implies $K = U$ .

Historical question references

Jones normalization and coefficient notation

Crossing number

Roberts, Knots Knotes, §4.4.8

Remark

Roberts's 1999 notes record the implication as a conjecture. Carmi and Cohen's arXiv:2606.22410v1, submitted on 2026-06-21, claims that every nontrivial classical knot has a nonzero coefficient vector $f_{q} (K)$ for some integer $q$ with $2 \leq q \leq 3 c (K)$ , and claims $V_{K} (x) = 1 \Rightarrow K = U$ . This scaffold has read the stated result and the paper's proof route but has not independently validated the claimed proof; it asserts neither proof acceptance nor a definitive current open status.

The source's machine argument still requires an independent diagram-level audit of Proposition 5.12, particularly its exceptional-state collapse, smoothing proxy, and two-machine locality. Its numerical estimate $B_{a u x} \leq B_{h o s t} + 12$ is consistent with the stated overhead $κ_{m a c h} = 11$ : the argument uses the next layer $r + 1$ after $r > B_{h o s t} + 11$ . This arithmetic check does not verify the geometric reductions. No Jones-detection theorem is registered by this scaffold.

Remark references

Carmi and Cohen, On trivial Jones–Vassiliev polynomials, §§1.3, 3, 5.6, 6.2–6.3

Roberts, Knots Knotes, §4.4.8

Kauffman, Combinatorial Knot Theory and the Jones Polynomial, §2

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Avishy Carmi and Eliahu Cohen, On trivial Jones–Vassiliev polynomials, arXiv:2606.22410v1, submitted 2026-06-21 (standard reference, not scraped)
Justin Roberts, Knots Knotes, §4.4.8 (standard reference, not scraped)
Louis H. Kauffman, Combinatorial Knot Theory and the Jones Polynomial, arXiv:2204.12104v2, §2 (standard reference, not scraped)