CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 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 framed degree-one weight system does not pass through 1T

Statement refuted

Assume countable choice. The framed self-linking number of a framed oriented knot is the linking number of the knot with a sufficiently small push-off along its framing. Let $D_{1}$ be the degree-one chord diagram consisting of an oriented circle with one chord. The framed self-linking invariant has a degree-one weight system $w$ with $w (D_{1}) = 2$ . Since $D_{1}$ is isolated, the unframed 1T quotient sets it to zero. Therefore $w$ does not factor through the unframed 4T+1T quotient. This counterexample distinguishes the framed and unframed diagram targets and makes no claim about separating knot types.

The diagram shows the isolated one-chord generator; the circle orientation fixes its cyclic skeleton and no proof relies on the picture.

Statement references

Framed 4T and unframed 4T+1T quotients

Framed self-linking symbol on one chord

Bar-Natan and Stoimenow, §1.1, PDF pp. 3–5

Chmutov, Duzhin, and Mostovoy, §§3.4 and 4.1–4.2, printed pp. 70–75 and 89–94

Facts & Assumptions

Given: Countable choice, the degree-one framed chord module, and its unframed quotient.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice is the hypothesis used for the framed normal-bundle branch.

[F1]

Framed chord diagrams and four-term relations: the framed module imposes 4T; the unframed module additionally sets every isolated chord to zero.

[F2]

Vassiliev invariants induce weight systems: under countable choice, framed self-linking has type at most one and its degree-one derivative is a framed weight system taking value $2$ on the one-chord diagram.

Counterexample

Given: As above.

1.1A1F1F2givenalgebra

The diagram $D_{1}$ has one chord and no other chords, so that chord is isolated. In degree one there is no 4T generator, since 4T has two chords. Thus [F2] gives a valid framed weight system with $w (D_{1}) = 2$ .

2.1F1F2step 1.1algebra∎

Every functional on the unframed quotient must vanish on $D_{1}$ , because [F1] imposes $D_{1} = 0$ . If $w$ factored through that quotient, it would satisfy $w (D_{1}) = 0$ , contrary to $w (D_{1}) = 2 \neq 0$ in $Z$ . Hence the framed weight system does not descend through 1T.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

The Fundamental Theorem of Vassiliev Invariants, §1.1 (standard reference, not scraped)
Introduction to Vassiliev Knot Invariants, §§3.4 and 4.1–4.2 (standard reference, not scraped)