TheoremStatement: Literature-sourcedProof: 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.

Linking number is invariant and writhe changes under a kink

Statement

Assume the Axiom of Countable Choice. For any two regular oriented diagrams of ambient-isotopic two-component links, the algorithmic linking numbers agree. Creating a positive type-I kink increases total writhe by $1$ and deleting it decreases writhe by $1$ ; creating a negative kink decreases writhe by $1$ and deleting it increases writhe by $1$ . Types II and III leave total writhe unchanged. For a knot component, blackboard self-linking changes by the same amount under a type-I move. Consequently writhe and blackboard self-linking are not invariants of unframed ambient isotopy, although the integer linking number of two distinct components is.

Statement references

Oriented Reidemeister theorem

Writhe and crossing sign

Blackboard self-linking calculation

Countable-choice hypothesis

Facts & Assumptions

Given: The Axiom of Countable Choice, regular oriented link diagrams, and the indicated ambient-isotopy relations.

[A1]

The Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Under ${A C}_{ω}$ , ambient-isotopic regular oriented link diagrams are connected by a finite sequence of planar isotopies and oriented Reidemeister moves (Reidemeister's theorem for oriented diagrams).

[F2]

A type-I move adds or removes a self-crossing; type II adds or removes two crossings of opposite signs; type III preserves the signs of its three crossings (Oriented Reidemeister moves).

[F3]

Writhe is the signed sum over all crossings, and self-writhe is the signed sum over one component's self-crossings (Oriented crossing signs, diagram writhe, and blackboard framing).

[F4]

Linking number is the integer half-sum of mixed crossing signs, invariant under all three Reidemeister moves (The mixed crossing sum is even and defines linking number).

[F5]

For a knot, blackboard self-linking equals diagram writhe (Blackboard self-linking equals diagram self-writhe).

Proof

1.1A1F1F4given

Apply [F1] to two regular diagrams of the same oriented link. By [F4], the integer linking number is unchanged by planar isotopy and by every move in the resulting finite sequence. Hence the two diagrams have equal linking number.

1.2F2F3algebra

A type-I move creates or deletes exactly one self-crossing and no mixed crossing by [F2]. Its contribution to the writhe in [F3] is therefore the sign $+ 1$ or $- 1$ of that crossing. A type-II move either contributes no mixed crossings or creates a mixed pair of opposite signs; in both cases its total writhe contribution is zero. A type-III move preserves each crossing sign, so it preserves writhe.

2.1A1F5step 1.2

By [F5], the knot component's blackboard self-linking is equal to the writhe of its diagram before and after each move. Thus a type-I move changes that self-linking by the same $+ 1$ or $- 1$ as writhe, while type II and III preserve it.

3.1A1F1F3step 1.2step 2.1construct∎

Let $D_{0}$ be a crossing-free diagram of the unknot, so $wr (D_{0}) = 0$ . Add a positive type-I kink to obtain $D_{1}$ . By [F1], the two diagrams represent ambient-isotopic unframed knots. The calculation in step 1.2 gives $wr (D_{1}) = 1$ . Hence writhe is not an invariant of unframed ambient isotopy; step 2.1 gives the same change for the blackboard self-linking.

Depends on

Used by

Writhe is not an invariant of an unframed knot Counterexample
Hopf and Whitehead link linking-number computations Example
The degree-one derivative of linking number Example

Dependency tree · two levels

38 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

Danny Calegari, Introduction to Knot Theory, University of Chicago course notes (2025), §3.3, pp. 26–29 (standard reference, not scraped)
Mohamed Elhamdadi, Mustafa Hajij, and Kyle Istvan, Framed Knots, arXiv:1910.10257v6 (2020), §3, pp. 3–9 (standard reference, not scraped)