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

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The normalized bracket is an oriented link invariant

Statement

Assume the Axiom of Countable Choice. Let $A$ be an indeterminate and let $D$ be a nonempty regular oriented link diagram in an oriented $3$ -sphere. Fix a unit viewing normal $n$ and orient the projection plane $P$ so $(u, v)$ is positive exactly when $(n, u, v)$ is positive in the ambient $3$ -space. At each crossing define $sgn (c) = sgn \det_{P} (v_{o v e r}, v_{u n d e r})$ from the oriented projected tangents and set $wr (D) = \sum_{c} sgn (c)$ . At each crossing use the local source chart in which the over-strand runs southwest to northeast; the A-smoothing joins the left endpoints and the right endpoints with weight $A$ , and the B-smoothing joins the top endpoints and the bottom endpoints with weight $A^{- 1}$ . Labels are transported by orientation-preserving changes of local coordinates. Let $C (D)$ be the finite crossing set. A state is a map $S : C (D) \to {A, B}$ ; let $a (S)$ and $b (S)$ count its A- and B-smoothings, and let $∣ S ∣$ be the number of circles after all crossings are smoothed. Set $⟨ D ⟩ = \sum_{S : C (D) \to {A, B}} A^{a (S) - b (S)} (- A^{2} - A^{- 2})^{∣ S ∣ - 1} .$ Define $f_{D} (A) = (- A^{3})^{- wr (D)} ⟨ D ⟩,$ where $wr (D)$ is the sum of the determinant signs of its oriented crossings. If two such diagrams represent ambient-isotopic oriented links, their $f_{D} (A)$ values are equal. Thus $f_{D}$ defines an invariant of nonempty oriented links.

Statement references

Kauffman bracket state sum

Crossing sign and writhe

Oriented Reidemeister theorem

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

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

Roberts, Knots Knotes (1999), §§4.1–4.4

Funar, A Crash Course in Quantum Invariants of Knots and 3-Manifolds, Chapter 3, §§1.1–2.3

Facts & Assumptions

Given: The Axiom of Countable Choice and two nonempty regular oriented diagrams of ambient-isotopic oriented links.

[A1]

The Axiom of Countable Choice is the assumption ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The normalized bracket is unchanged by planar isotopy and every oriented R1, R2, or R3 move (Kauffman bracket curl and writhe correction).

[F2]

Under ${A C}_{ω}$ , two 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).

[F3]

Every oriented link may be moved off the point at infinity into $R^{3}$ , and the directions giving regular projections are dense in $S^{2}$ (Oriented links in the three-sphere and ambient isotopy, Existence of regular projections).

Proof

1.1F1induction

By [F1], each planar isotopy and each oriented Reidemeister move in a finite sequence preserves the normalized bracket value. Repeated equality along a finite sequence therefore preserves $f_{D} (A)$ .

2.1A1F2F3givenstep 1.1∎

By [A1] and [F2], the two given diagrams are joined by such a finite sequence. Applying step 1.1 to that sequence gives equal normalized bracket values. By [F3], every nonempty oriented link has a regular diagram. Hence $f_{D}$ defines a value on every nonempty oriented link type.

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, printed pp. 430–432 and 466–469 (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, §§4.1–4.4, printed pp. 21–26 (standard reference, not scraped)
François Funar, A Crash Course in Quantum Invariants of Knots and 3-Manifolds, Chapter 3, §§1.1–2.3 (standard reference, not scraped)