TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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The bracket state sum matches the library Jones specialization

Statement

Assume the Axiom of Choice. Let $A$ and $t$ be formal variables and $K$ a nonempty oriented link, and let $D$ be any regular oriented diagram representing $K$ . At each crossing of $D$ use the source chart with over-strand southwest-to-northeast: A joins left and right endpoints with coefficient $A$ , and B joins top and bottom endpoints with coefficient $A^{- 1}$ . A state $S : C (D) \to {A, B}$ assigns one smoothing at every crossing; let $a (S)$ and $b (S)$ count these choices and $∣ S ∣$ count the resulting circles. 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 oriented $3$ -sphere. Define $sgn (c) = sgn \det_{P} (v_{o v e r}, v_{u n d e r}), wr (D) = \sum_{c} sgn (c),$ $f_{K} (A) = (- A^{3})^{- wr (D)} \sum_{S : C (D) \to {A, B}} A^{a (S) - b (S)} (- A^{2} - A^{- 2})^{∣ S ∣ - 1} .$ This value is independent of the chosen regular oriented diagram of $K$ . For an $n$ -strand braid $β$ whose oriented closure is $K$ , let $e : B_{n} \to Z$ be the homomorphism $e (σ_{i}) = 1$ , where $σ_{i}$ is the positive geometric Artin half twist. Let $H (n)$ be the type-A Hecke algebra with generators $T_{i}$ , quadratic relation $T_{i}^{2} = (v - 1) T_{i} + v$ , braid relations, and distant commutation, and let $π_{n} (σ_{i}) = T_{i}$ . Its normalized Markov trace ${tr}_{n}$ satisfies ${tr}_{n} (1) = 1$ , cyclicity, tower restriction, and ${tr}_{n + 1} (x T_{n}) = z {tr}_{n} (x)$ . Define the library Jones value by $V_{K} (t) = s^{e (β)} {(- \frac{s^{2} + 1}{s})}^{n - 1} {tr}_{n} (π_{n} (β)) ∣_{v = s^{2}, z = - 1 / (s^{2} + 1), t = s^{2}},$ where $s = t^{1 / 2}$ . With $A = t^{- 1 / 4}$ , these values satisfy $V_{K} (t) = f_{K} (t^{- 1 / 4}) .$

Statement references

Hecke-trace Jones specialization

Normalized bracket invariance

Braid closure trace dictionary

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: Full AC, a nonempty oriented link $K$ , a braid closure of $K$ , and the two polynomial constructions in the statement.

[A1]

Full AC means that every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

Full AC implies countable choice (AC implies DC implies countable choice).

[F2]

The positive Artin half twist has determinant crossing sign $+ 1$ ; the opposite reflected half twist exchanges over and under and has sign $- 1$ (Oriented crossing signs, diagram writhe, and blackboard framing, The elementary geometric half twist, its support disc, and its opposite).

[F3]

The braid exponent sum is additive, with $e (σ_{i}) = 1$ and $e (σ_{i}^{- 1}) = - 1$ (The exponent sum of a braid).

[F4]

For every braid closure, the normalized bracket satisfies $f_{\hat{β}} (A) = (- A^{3})^{- wr (\hat{β})} ⟨ \hat{β} ⟩,$ and its value is independent of the regular diagram under AC $_{ω}$ (The normalized bracket is an oriented link invariant).

[F5]

For every $n$ -braid, the raw bracket has the trace expression $⟨ \hat{β} ⟩ = δ^{n - 1} (- A)^{e (β)} {tr}_{n} (π_{n} (β)) ∣_{v = A^{- 4}, z = - 1 / (A^{- 4} + 1)}, δ = - A^{2} - A^{- 2}$ (Hecke–bracket trace dictionary and local descent).

[F6]

The library Jones value is invariant under braid equivalence and Markov moves under full AC, and is given by the trace formula in the Statement (The Hecke trace construction is an oriented link invariant, The Temperley-Lieb quotient and the Jones specialization).

[F7]

Every nonempty oriented link has a braid closure under full AC (Alexander's theorem: every link is a closed braid, The closure of a geometric braid).

[F8]

Every element of $B_{n}$ is represented by a finite word in the Artin generators and their inverses; each letter is one geometric half twist or its opposite, and the closure joins the braid endpoints (The braid group by Artin presentation, The elementary geometric half twist, its support disc, and its opposite, The closure of a geometric braid).

Proof

1.1F2F3F8algebra

A closed braid diagram has one crossing for each letter in a finite Artin word and no crossings at its closure arcs. Each positive generator contributes $+ 1$ to writhe by [F2]. In the reflected inverse half twist, the over- and under-strands exchange while their oriented projected tangent vectors retain their respective directions; exchanging the two vectors negates their determinant, so each inverse generator contributes $- 1$ . Summing over the word and using the homomorphism in [F3] and the finite word representation in [F8] gives $wr (\hat{β}) = e (β) .$

2.1F4F5F6step 1.1algebra

Put $q = A^{- 2}$ and $t = q^{2}$ . Then $δ = - A^{2} - A^{- 2} = - (q + q^{- 1}) = - (t + 1) / q$ . By [F5] and step 1.1, $\begin{aligned} f_{\hat{β}} (A) & = (- A^{3})^{- e (β)} δ^{n - 1} (- A)^{e (β)} {tr}_{n} (π_{n} (β)) \\ = q^{e (β)} δ^{n - 1} {tr}_{n} (π_{n} (β)) \\ = q^{e (β)} {(- \frac{q^{2} + 1}{q})}^{n - 1} {tr}_{n} (π_{n} (β)) . \end{aligned}$ This is exactly the library trace normalization with $s = q$ and $t = s^{2}$ from [F6]. Thus the two constructions agree on every braid closure.

3.1F5F6step 2.1algebra

Put $R_{A} = Z [A^{\pm 1}, (A^{- 4} + 1)^{- 1}]$ , as in [F5]. The target coefficient ring in [F6] is $T = Z [s^{\pm 1}, (s^{2} + 1)^{- 1}]$ . The substitution $s \mapsto A^{- 2}$ , $t = s^{2} \mapsto A^{- 4}$ defines a ring map $T \to R_{A}$ . It is injective because distinct Laurent powers $s^{k}$ map to distinct Laurent monomials $A^{- 2 k}$ , and the localization denominator maps to the unit $A^{- 4} + 1$ . Hence the equality from step 2.1 is an equality in $T$ , with $t = s^{2}$ , rather than only after embedding the two values in a larger ring.

4.1A1F1F4F6F7step 2.1step 3.1given∎

By [A1] and [F1], full AC supplies the countable-choice hypothesis of [F4]. By [F7], choose a braid closure for the given nonempty link $K$ . The calculation in step 2.1 and its ring identification in step 3.1 prove equality on that closure; the invariance in [F4] and [F6] makes both values independent of the chosen regular diagram and braid representative. Therefore $V_{K} (t) = f_{K} (t^{- 1 / 4})$ for every nonempty oriented link.

Depends on

Used by

A nugatory kink defeats crossing minimality Counterexample
Jones–Vassiliev coefficient vectors Definition
A trefoil Jones–Vassiliev coefficient calculation Example
Trefoil and figure-eight Jones spans Example
Trefoil, mirror, and unknot bracket calculations Example
Unknot, trivial color, and a three-crossing braid value Example
Adequacy and Jones span of reduced alternating diagrams Lemma
Reduced and unreduced Khovanov Euler characteristics and the Jones polynomial Theorem

Dependency tree · two levels

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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)