Pipeline-generated

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.

✓ 1 result · all verified · 0 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Vassiliev invariants and chord diagrams — Examples and Counterexamples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Writhe is not an unframed knot invariant

Statement refuted

For every oriented knot diagram $D$ , the writhe $w (D) = \sum_{c} sgn (c)$ , with $sgn (c) = sgn \det (v_{o v e r}, v_{u n d e r})$ in the oriented projection plane with positive viewing normal, is unchanged by unframed Reidemeister moves and therefore defines an unframed finite-type invariant of type at most $1$ .

Statement references

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

Chmutov, Duzhin, and Mostovoy, §4.2.2, pp. 92–93

Diagram

The picture is a local Reidemeister I pair: the vertical branch passes over the horizontal branch in the one-crossing curl, while the rest of the knot is unchanged.

At the single crossing of $D_{1}$ , the vertical over-tangent points down and the horizontal under-tangent points right, so the determinant convention gives sign $+ 1$ .

Facts & Assumptions

Given: The two oriented diagrams in the counterexample below, and the sign convention in the refuted statement.

[F1]

Each oriented Reidemeister move is realized by an ambient isotopy: each local oriented Reidemeister I move is realized by an ambient isotopy of the represented oriented link.

[F2]

Singular knots and Vassiliev derivatives: a finite-type invariant must first be an invariant of ambient-isotopy classes of oriented knots.

[F3]

Oriented crossing signs, diagram writhe, and blackboard framing: writhe is the sum of the crossing signs, with the determinant convention stated in the refuted statement.

Counterexample

Let $D_{0}$ be a crossing-free diagram of the oriented unknot and let $D_{1}$ be obtained from it by inserting one positive Reidemeister I curl. Both diagrams represent the same oriented unknot. The writhe values are $w (D_{0}) = 0$ and $w (D_{1}) = 1$ .

1.1givenF1

By [F1], $D_{0}$ and $D_{1}$ represent the same unframed oriented knot, since they differ by one oriented Reidemeister I move.

2.1givenF2F3step 1.1algebra∎

The diagram $D_{0}$ has no crossings, so $w (D_{0}) = 0$ . The diagram $D_{1}$ adds exactly one positive crossing and leaves all other crossings absent, so $w (D_{1}) = + 1$ by [F3]. Therefore writhe is not constant on the ambient-isotopy class of the unknot; by [F2] it cannot be an unframed finite-type knot invariant.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedverified 2026-10-22 (gpt-6.1-sol)Open item page →

The degree-one derivative of linking number

Example

Assume ${A C}_{ω}$ . For an oriented two-component link $L = L_{1} \cup L_{2}$ , let $lk (L_{1}, L_{2})$ be one half the sum of the signs of all crossings between distinct components, using $sgn \det (v_{o v e r}, v_{u n d e r})$ in the oriented projection plane with positive viewing normal. Extend the singular finite-difference formula to links by resolving marked crossings independently. At one marked mixed crossing, the positive-minus-negative derivative of $lk$ is $+ 1$ ; reversing the order of those two resolutions gives $- 1$ . A marked self-crossing has derivative $0$ . At any two distinct marked crossings, mixed or self, the second derivative is $0$ , so linking number is a type-at-most-one invariant of two-component links and, by the nonzero first derivative, has exact type one. The total pairwise linking number of a one-component knot is $0$ , because the defining sum over distinct component pairs is empty.

Example references

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

Chmutov, Duzhin, and Mostovoy, §§2.2.1 and 3.2.2, pp. 33–34 and 86

Facts & Assumptions

Given: The crossing-sign convention and the two-component oriented-link linking-number formula in the Example statement. The topological invariance of this quantity uses the separate ${A C}_{ω}$ supplier branch.

[F1]

Singular knots and Vassiliev derivatives: link finite differences are the alternating sums over independent marked-crossing resolutions.

[F2]

Oriented crossing signs, diagram writhe, and blackboard framing: a crossing sign is $sgn \det (v_{o v e r}, v_{u n d e r})$ ; a positive crossing has sign $+ 1$ and a negative crossing has sign $- 1$ .

[F3]

Compute oriented linking number from a diagram: on a two-component diagram the linking-number algorithm returns one half of the signed sum of mixed crossings.

[F4]

The diagram linking-number scan terminates and returns the stated integer: under ${A C}_{ω}$ , the finite diagram scan terminates and returns the integer linking number.

[F5]

Linking number is invariant and writhe changes under a kink: under ${A C}_{ω}$ , the algorithmic linking number is unchanged by oriented Reidemeister moves and hence is invariant under ambient isotopy.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ asserts that every sequence of nonempty sets has a choice function; only the topological supplier chain [F4]–[F5] uses it.

Verification

1.1givenA1F1F2F3F4F5algebra

Let $C$ be half the signed sum of all mixed crossings except the marked one. Under [A1], [F4] identifies the diagram half-sum with the integer linking number and [F5] makes it an ambient-isotopy invariant. By [F2] and [F3], the positive and negative resolutions contribute respectively $+ \frac{1}{2}$ and $- \frac{1}{2}$ to this linking number, so $lk (L_{1}, L_{2})_{+} - lk (L_{1}, L_{2})_{-} = (C + \frac{1}{2}) - (C - \frac{1}{2}) = 1$ . This is the stated positive-minus-negative derivative under [F1]; reversing that subtraction changes its sign.

1.2givenalgebra

A knot has one component and therefore no unordered pair of distinct components; the total pairwise linking sum contains no terms and equals $0$ .

2.1givenA1F1F2F3F4step 1.1algebra∎

If both marked crossings are mixed, write their resolution signs as $ϵ_{1}, ϵ_{2} \in {+ 1, - 1}$ and absorb every other mixed crossing into $C$ . By [F2]–[F4], $lk (L_{ϵ_{1}, ϵ_{2}}) = C + \frac{1}{2} (ϵ_{1} + ϵ_{2})$ . The alternating second difference is zero: the constant has coefficient $1 - 1 - 1 + 1 = 0$ , and each term linear in one $ϵ_{i}$ cancels when the other sign is summed. If either marked crossing is a self-crossing, the link formula [F3] is independent of that resolution, so the alternating sum over its two choices cancels term by term. Thus $D_{2} = 0$ for every pair of marked crossings by [F1].

Vassiliev invariants and chord diagrams — Examples and Counterexamples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

Writhe is not an unframed knot invariant

Statement refuted

Statement references

Diagram

Facts & Assumptions

Counterexample

The degree-one derivative of linking number

Example

Example references

Facts & Assumptions

Verification

Sources