CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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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.
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A type-I kink changes the raw bracket

Statement refuted

The false statement is: let $A$ be an indeterminate and, for every nonempty regular oriented diagram $D$ of an unframed knot with finite crossing set $C (D)$ , let $⟨ D ⟩ = \sum_{S : C (D) \to {A, B}} A^{a (S) - b (S)} (- A^{2} - A^{- 2})^{∣ S ∣ - 1},$ where $a (S)$ and $b (S)$ count A- and B-smoothings, the over-strand runs southwest to northeast, A joins the left and right endpoint pairs with weight $A$ , B joins the top and bottom endpoint pairs with weight $A^{- 1}$ , and $∣ S ∣$ is the number of state circles. Then $⟨ D ⟩$ depends only on the ambient-isotopy class of the represented oriented knot.

Statement references

Kauffman bracket state sum

Type-I curl factors

Oriented type-I move

Kauffman, An Invariant of Regular Isotopy (1990), §§III and VI

Kauffman, Combinatorial Knot Theory and the Jones Polynomial (2022), §2

Roberts, Knots Knotes (1999), §§4.1 and 4.3

Funar, A Crash Course in Quantum Invariants of Knots and 3-Manifolds, Chapter 3, §1.1

Facts & Assumptions

Given: The false statement above and the bracket state sum on nonempty regular oriented diagrams after their strand arrows are forgotten.

[F1]

A crossing-free one-component diagram has one state with one circle and bracket $(- A^{2} - A^{- 2})^{1 - 1} = 1$ (Kauffman bracket smoothing and loop value).

[F2]

Adding a positive type-I kink multiplies the raw bracket by $- A^{3}$ (Kauffman bracket curl and writhe correction).

[F3]

An oriented type-I kink is an allowed local Reidemeister replacement and each such local replacement is realized by an ambient isotopy of the oriented link (Oriented Reidemeister moves, Each oriented Reidemeister move is realized by an ambient isotopy).

Counterexample

1.1F1construct

Let $D_{0}$ be a crossing-free diagram of the oriented unknot. By [F1], $⟨ D_{0} ⟩ = 1$ .

2.1F2F3algebra∎

Obtain $D_{1}$ by adding one positive type-I kink to $D_{0}$ . By [F3] the two diagrams represent ambient-isotopic unframed oriented knots, while [F2] gives $⟨ D_{1} ⟩ = - A^{3}$ . Since $A$ is an indeterminate, $- A^{3} \neq 1$ in $Z [A^{\pm 1}]$ . Thus the proposed value differs on two diagrams of the same unframed knot, refuting the statement.

Depends on

Used by

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Sources

Louis H. Kauffman, An Invariant of Regular Isotopy, Transactions of the AMS 318(2) (1990), §§III and VI (standard reference, not scraped)
Louis H. Kauffman, Combinatorial Knot Theory and the Jones Polynomial, arXiv:2204.12104v2, §2 (standard reference, not scraped)
Justin Roberts, Knots Knotes, §§4.1 and 4.3 (standard reference, not scraped)
François Funar, A Crash Course in Quantum Invariants of Knots and 3-Manifolds, Chapter 3, §1.1 (standard reference, not scraped)