CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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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.

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Writhe is not an invariant of an unframed knot

Statement refuted

The false statement is: for every regular oriented diagram $D$ of an unframed knot, the writhe $wr (D)$ , defined as the sum of its oriented crossing signs, depends only on the ambient-isotopy class of the represented knot.

Statement references

Crossing sign and writhe definitions

Ambient isotopy of oriented knots

Diagram

The right-hand curve is one oriented circle with a single local kink. At $c$ the over tangent is $e_{1}$ and the under tangent is $e_{2}$ in the positive projection chart, so its crossing sign is $\det (e_{1}, e_{2}) = + 1$ . The lower-right loop is the added type-I kink; deleting it gives the crossing-free unknot diagram on the left. The figure illustrates the construction; the proof uses the Reidemeister theorem and the signed-sum definition.

Facts & Assumptions

Given: The Axiom of Countable Choice and the false statement above.

[A1]

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

[F1]

A regular oriented diagram represents a smooth oriented link, and an oriented Reidemeister type-I move adds or removes a self-crossing kink (Regular oriented link diagrams, Oriented Reidemeister moves).

[F2]

Under ${A C}_{ω}$ , two regular diagrams connected by an oriented Reidemeister move represent ambient-isotopic oriented links (Reidemeister's theorem for oriented diagrams).

[F3]

The writhe is the sum of crossing signs; a positive type-I crossing contributes $+ 1$ (Oriented crossing signs, diagram writhe, and blackboard framing).

[F4]

Blackboard self-linking equals diagram writhe (Blackboard self-linking equals diagram self-writhe).

Counterexample

1.1F1construct

Let $D_{0}$ be a crossing-free diagram of the oriented unknot and let $D_{1}$ be obtained by one positive type-I move. By [F1] the new crossing is a self-crossing and is positive.

1.2A1F2F3algebra

By [F2], the type-I move is realized by an ambient isotopy, so $D_{0}$ and $D_{1}$ represent the same unframed oriented knot. By [F3], $wr (D_{0}) = 0$ while $wr (D_{1}) = 1$ . Therefore the displayed false statement fails.

2.1F4step 1.1step 1.2∎

By [F4], the blackboard self-linking values for these two diagrams are also $0$ and $1$ , respectively. This is the framing change carried by the single positive kink; it does not contradict equality of knot types after forgetting the framing.

Depends on

Used by

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Dependency tree · two levels

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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)