CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

The unknot defeats a universal coefficient-bound extension

Statement refuted

Assume full AC for the Jones invariant and ${A C}_{ω}$ for $c (K)$ . A classical knot is a smooth embedding $S^{1} ↪ S^{3}$ considered up to ambient isotopy. A regular diagram is a generic plane projection with only transverse double points, each marked with over/under information. The false universal extension is: for every classical knot $K$ , including the unknot, there is an integer $q$ with $2 \leq q \leq 3 c (K)$ and $f_{q} (K) \neq (0, 0)^{T}$ . Here $V_{K} (x)$ is the normalized library Jones polynomial with $V_{U} (x) = 1$ , $f_{q} (K) = (a_{q} (K), b_{q} (K))^{T}$ is defined by the unique expansion $V_{K} (x) = \sum_{q \geq 0} (a_{q} (K) + b_{q} (K) x) (x - x^{- 1})^{q}$ , and $c (K)$ is the least crossing number of a regular diagram of $K$ .

The boundary input is the unknot $U$ , represented by the crossing-free circle:

Statement references

Jones–Vassiliev coefficient vectors

Crossing number

Carmi and Cohen, On trivial Jones–Vassiliev polynomials, Theorem 6.8

Facts & Assumptions

Given: Full AC, ${A C}_{ω}$ , and the unknot $U$ with its one-component crossing-free diagram.

[F1]

The library Jones polynomial has $V_{U} (x) = 1$ , and the coefficient vector is obtained from the unique finite expansion in powers of $p = x - x^{- 1}$ (Jones–Vassiliev coefficient vectors).

[F2]

Under ${A C}_{ω}$ , $c (K)$ is the least element of the set of crossing counts of regular diagrams representing $K$ (Crossing number of a link).

[A1]

Full AC is the family-choice assertion (The Axiom of Choice).

[A2]

${A C}_{ω}$ selects an element from each countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Counterexample

1.1givenA1F1constructalgebra

Under [A1], $V_{U} (x) = 1$ by [F1]. Its unique coefficient expansion is $a_{0} (U) = 1$ , $b_{0} (U) = 0$ , and $a_{q} (U) = b_{q} (U) = 0$ for every $q \geq 1$ . Thus $f_{q} (U) = (0, 0)^{T}$ for every positive order.

1.2givenA2F2algebra

The displayed diagram has zero crossings, so under [A2] the set in [F2] contains $0$ ; all crossing counts are nonnegative. Hence $c (U) = 0$ , and there is no integer $q$ satisfying $2 \leq q \leq 3 c (U) = 0$ .

2.1givenstep 1.1step 1.2∎

The proposed universal statement includes $U$ , but steps 1.1 and 1.2 show that no required nonzero vector or admissible order exists for it. Therefore the statement is false.

Remarks

The source's actual coefficient claim assumes that $K$ is nontrivial, so this counterexample refutes only the broader universal extension. The presence or absence of DOI or journal metadata is bibliographic information; by itself it proves neither the source theorem nor its negation.

Remarks references

Carmi and Cohen, On trivial Jones–Vassiliev polynomials, Theorem 6.8

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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)