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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Normalized colored Jones sequence

Definition

Let $A$ be an indeterminate, put $q = A^{4}$ , $q_{K} = A^{2}$ , and $q_{L} = A^{- 2} = q_{K}^{- 1}$ , and work over $F = Q (A)$ . An oriented knot in $S^{3}$ means a smooth embedded circle with an orientation, considered up to orientation-preserving ambient isotopy. For an integer $r$ , put $[r] = \frac{q_{K}^{r} - q_{K}^{- r}}{q_{K} - q_{K}^{- 1}}, [m]! = \prod_{r = 1}^{m} [r] (m \geq 1), [0]! = 1.$ For each $N \geq 1$ , put $q_{N} = \exp (2 π i / N)$ . For each integer $N \geq 1$ , let $V_{N}$ have basis $v_{0}, \dots, v_{N - 1}$ and define the $U_{q_{L}} ({s l}_{2})$ action in the library coproduct convention by $K_{L} v_{j} = q_{L}^{N - 1 - 2 j} v_{j}, E v_{j} = [N - j] v_{j - 1}, F v_{j} = [j + 1] v_{j + 1},$ where $v_{- 1} = v_{N} = 0$ . In the universal $R$ -matrix formula use $q_{K}$ and $H v_{j} = (N - 1 - 2 j) v_{j}$ ; on $V_{N} \otimes V_{N}$ its action is the finite sum $R_{N} = q_{K}^{H \otimes H / 2} \sum_{r = 0}^{N - 1} q_{K}^{r (r - 1) / 2} \frac{(q_{K} - q_{K}^{- 1})^{r}}{[r]!} E^{r} \otimes F^{r},$ and the positive crossing is $c_{N} = P R_{N}$ , where $P (x \otimes y) = y \otimes x$ . This is an irreducible module of dimension $N$ . Equip these matrices with the Kauffman-bracket half-ribbon data: the twist matrix is the scalar $θ_{N} = λ_{N} I$ with $λ_{N} = (- 1)^{N - 1} A^{N^{2} - 1}$ , and the matrix $G_{N} = (- 1)^{N - 1} diag (A^{2 (N - 1 - 2 j)})_{j = 0}^{N - 1}$ is the pivotal comparison used in the closure trace. The resulting generic quantum dimension is $d_{N} = Tr (G_{N}) = (- 1)^{N - 1} \frac{A^{2 N} - A^{- 2 N}}{A^{2} - A^{- 2}} .$ Use the contragredient left-dual action $(x \cdot f) (v) = f (S (x) v)$ on $V_{N}^{*}$ , with evaluation $ev (f \otimes v) = f (v)$ and coevaluation $1 \mapsto \sum_{j = 0}^{N - 1} v_{j} \otimes v_{j}^{*}$ . The ambient category is the $F$ -linear category generated by the modules $V_{N}$ for all $N \geq 1$ under finite tensor products, contragredient duals, finite direct sums and images of module-idempotents, whose morphisms are module homomorphisms, and whose tensor-product action is induced by the stated Hopf coproduct; its tensor unit is $F$ with the trivial action, so its tensor-unit endomorphism ring is $F$ . These matrix data satisfy the ribbon-category axioms: braiding and inverse braiding, both duality snakes, monoidal pivotal comparison, balancing and dual compatibility of twist. Assume the Axiom of Choice and Countable Choice for the framed-tangle and Markov constructions. An oriented knot $K \subset S^{3}$ presented as the closure of an $n$ -strand braid $β$ has generic normalized value $J_{N}^{g e n} (K; q) = λ_{N}^{- w (β)} \frac{Tr (G_{N}^{\otimes n} ρ_{N} (β))}{d_{N}} \in F,$ where, for a word $β = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{m}}^{ε_{m}}$ with $ε_{j} \in {1, - 1}$ , $w (β) : = \sum_{j = 1}^{m} ε_{j}$ , and $ρ_{N} (β)$ is obtained by replacing each positive braid generator by $c_{N}$ on the corresponding adjacent tensor factors and each inverse generator by $c_{N}^{- 1}$ , then composing in word order. This quotient is an integral Laurent polynomial in $A^{4}$ . Define the normalized colored Jones sequence by the unique $J_{N} (K; q) \in Z [q, q^{- 1}]$ satisfying $J_{N} (K; A^{4}) = J_{N}^{g e n} (K; A^{4})$ . Its root value $J_{N} (K; q_{N})$ is obtained by substituting into this Laurent polynomial. The variable transport to the ordinary Jones convention is $t = q^{- 1} = A^{- 4}$ .

Definition references

All-color ribbon identities ↗

Integral Laurent representative and safe root evaluation ↗

Drinfeld–Jimbo generators and relations

Library coproduct and antipode convention

Kauffman-bracket half-ribbon convention in dimension two

Colored framed-link evaluation

Scalar twist correction under Markov moves

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§5.6 and 8.3

Snyder and Tingley, The half-twist for $U_{q} (g)$ representations, §§3–5

Murakami, An Introduction to the Volume Conjecture, §2.4

Habiro, On the quantum ${s l}_{2}$ invariants of knots and integral homology spheres, §§2.3 and 3.3

Facts & Assumptions

Given: An integer $N \geq 1$ , the Axiom of Choice and Countable Choice, the generic module and matrices from the Definition, an oriented knot $K$ , and a braid word $β$ whose closure represents $K$ .

[F1]

The library presentation has $K_{L} E K_{L}^{- 1} = q_{L}^{2} E$ , $K_{L} F K_{L}^{- 1} = q_{L}^{- 2} F$ , and $E F - F E = (K_{L} - K_{L}^{- 1}) / (q_{L} - q_{L}^{- 1})$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F2]

The library coproduct is $Δ (E) = E \otimes K_{L}^{- 1} + 1 \otimes E$ and $Δ (F) = F \otimes 1 + K_{L} \otimes F$ (The Drinfeld–Jimbo formulas define a Hopf algebra).

[F3]

A ribbon category has a braiding, chosen duals, and a twist satisfying the balancing and dual-compatibility identities (Braiding, Left dual and right dual object, Twist and ribbon structure).

[F4]

For an absolutely simple color with scalar twist $λ I$ , the generic framed trace corrected by $λ^{- w (β)}$ is invariant under oriented unframed Markov equivalence, under the stated Axiom of Choice (Scalar twist correction removes the framing).

[F5]

The framed colored-link value is the ribbon-tangle evaluation in ${End}_{C} (1)$ (Colored framed-link evaluation from a ribbon category). For an $X$ -colored closed braid, its categorical value is $t_{n} (β) = {Tr}_{L} (j_{X^{\otimes n}} ρ_{n} (β))$ (The ribbon evaluation of an $X$ -colored closed braid) and equals the tangle evaluation of its blackboard-framed closure (The ribbon trace equals the framed-closure evaluation).

[F6]

Absolute simplicity means $End (V_{N}) = F I$ (Absolutely simple objects).

[F7]

The finite matrices satisfy all ribbon identities for every $N \geq 1$ and their pivotal trace of the identity is $d_{N}$ (All generic colored Jones matrices satisfy the ribbon identities ↗).

[F8]

The normalized generic quotient for every knot and color belongs to $Z [A^{4}, A^{- 4}]$ (The normalized generic colored Jones quotient is an integral Laurent polynomial ↗).

[A2]

The Axiom of Choice supplies a choice function for every family of nonempty sets (The Axiom of Choice).

[A3]

Countable choice supplies a choice function for every sequence of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: The hypotheses in the Definition.

1.1F1givenalgebra

The diagonal weights give $K_{L} E K_{L}^{- 1} v_{j} = q_{L}^{2} E v_{j}$ and $K_{L} F K_{L}^{- 1} v_{j} = q_{L}^{- 2} F v_{j}$ . For the commutator, $E F v_{j} - F E v_{j} = ([j + 1] [N - j - 1] - [N - j] [j]) v_{j} = [N - 1 - 2 j] v_{j}$ ; the last equality follows by expanding the two quantum-number products over $q_{K} - q_{K}^{- 1}$ . Since $[r]_{q_{K}} = [r]_{q_{L}}$ , this is $(K_{L} - K_{L}^{- 1}) (q_{L} - q_{L}^{- 1})^{- 1} v_{j}$ . The generators therefore satisfy the defining ${s l}_{2}$ relations.

1.2F6givenalgebra

The eigenvalues $q_{L}^{N - 1 - 2 j}$ of $K_{L}$ are pairwise distinct in $F$ . Any module endomorphism is therefore diagonal in the given basis; commuting with $E$ , whose successive coefficients $[N - j]$ are nonzero in $F$ , forces all diagonal entries to be equal. Thus ${End}_{U_{q_{L}} ({s l}_{2})} (V_{N}) = F I$ and the color is absolutely simple by [F6].

1.3givenalgebra

If $N = 1$ , $V_{N}$ is one-dimensional and irreducible. For $N > 1$ , let $W$ be a nonzero submodule and choose $x = \sum_{j} c_{j} v_{j} \in W$ nonzero. The eigenvalues of $K_{L}$ are distinct, so the polynomial spectral projection for any $j$ with $c_{j} \neq 0$ maps $x$ to $c_{j} v_{j}$ ; hence $v_{j} \in W$ . Repeated application of $E$ gives $v_{0} \in W$ , since every coefficient $[N - k]$ encountered is nonzero. Repeated application of $F$ to $v_{0}$ gives each $v_{k}$ because the coefficients $[k]$ are nonzero. Thus $W = V_{N}$ , proving irreducibility.

1.4F2F3F5F7givenalgebra

The displayed sum for $d_{N}$ is a geometric sum, so $d_{N} = (- 1)^{N - 1} [N]$ . Its numerator $A^{2 N} - A^{- 2 N}$ is a nonzero Laurent polynomial and its denominator $A^{2} - A^{- 2}$ is nonzero in $F$ ; hence the generic normalization is defined. The Hopf coproduct [F2] supplies the tensor-power action used in $ρ_{N} (β)$ . By the braiding, dual, and ribbon axioms [F3] and the all-color ribbon identities [F7], the comparison on $V_{N}^{\otimes n}$ is $G_{N}^{\otimes n}$ . The basis coevaluation and evaluation in the left categorical trace [F5] then give the ordinary matrix trace $Tr (G_{N}^{\otimes n} ρ_{N} (β))$ , which is the scalar of the framed closure.

2.1A2A3F4F6F7step 1.2step 1.4given

By [F4], [F7], and [A2], $λ_{N}^{- w (β)} t_{N} (β)$ is unchanged by every conjugation and Markov stabilization, since $V_{N}$ is absolutely simple and its specified twist scalar is nonzero. Countable choice [A3] supplies the evaluator's hypothesis. Dividing by the fixed nonzero $d_{N}$ preserves that equality, so the displayed generic value is independent of the braid representative.

2.2F5step 1.4givenalgebra

The unknot is represented by the one-strand identity braid, whose exponent sum is zero and whose pivotal trace is $Tr (G_{N}) = d_{N}$ . Hence $J_{N}^{g e n} (◯; q) = 1$ . For $N \geq 2$ , at $q_{N} = e^{2 π i / N}$ choose $A_{N} = e^{π i / (2 N)}$ ; then $A_{N}^{4} = q_{N}$ and $A_{N}^{2} - A_{N}^{- 2} \neq 0$ , while $A_{N}^{2 N} - A_{N}^{- 2 N} = e^{π i} - e^{- π i} = 0$ . Thus $d_{N} (q_{N}) = 0$ : the generic quotient cannot be evaluated there by dividing by $d_{N}$ .

3.1A2A3F8step 2.1step 2.2givenalgebra∎

Apply [F8] to the given knot-closing braid and color under [A2] and [A3]. The generic quotient therefore has a unique integral Laurent representative in $q = A^{4}$ . Substitution into that finite Laurent sum is defined for every nonzero complex $q$ , including $q_{N}$ ; it does not divide by the specialized quantum dimension. Together with steps 2.1 and 2.2, this proves the stated representative independence, unknot normalization and root-evaluation convention.

Remarks

The all-color ribbon identities are proved by the finite-series interpolation argument in the first well-definedness supplier. The second proves integral Laurentness by cutting the knot into an integral scalar endomorphism and then proving invariance under $A \mapsto i A$ . Thus root evaluation uses the resulting Laurent polynomial, even though the generic quantum dimension vanishes at the specified root for every $N \geq 2$ .

Depends on

Used by

Dependency tree · two levels

50 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, §§5.6, 8.3, 8.9–8.10 (standard reference, not scraped)
Noah Snyder and Peter Tingley, The half-twist for U_q(g) representations, §§3–5 (standard reference, not scraped)
Hitoshi Murakami, An Introduction to the Volume Conjecture, §2.4 (standard reference, not scraped)
Kazuo Habiro, On the quantum sl2 invariants of knots and integral homology spheres, §§2.3 and 3.3 (standard reference, not scraped)