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

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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.

Depends on

Used by

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Dependency tree · two levels

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Sources

The Fundamental Theorem of Vassiliev Invariants (standard reference, not scraped)
Introduction to Vassiliev Knot Invariants (standard reference, not scraped)