LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The normalized closure trace is cyclic and satisfies the Markov rules

Statement

For $δ \in C^{\times}$ , the maps $τ_{n}$ of Normalized Temperley–Lieb trace from diagram closure satisfy

$τ_{n} (x y) = τ_{n} (y x), τ_{n + 1} (x \otimes 1) = τ_{n} (x), τ_{n + 1} ((x \otimes 1) d_{n}) = δ^{- 1} τ_{n} (x)$

for every $n \geq 0$ and $x, y \in {T L}_{n} (C, δ)$ in the first two identities, and for every $n \geq 1$ and $x \in {T L}_{n} (C, δ)$ in the third identity, where $d_{n}$ is the elementary cup-cap diagram on the last two strands.

Facts & Assumptions

Given: A nonzero loop value $δ$ , diagram closure, and stacking composition.

[F1]

Stacking is associative, each closed component contributes $δ$ , and the elementary generator $d_{n}$ is a cap and cup on the last two positions (Temperley–Lieb planar diagram data and endomorphism modules, Stacking is associative and horizontal juxtaposition is strict monoidal, The elementary diagram generators satisfy the local Temperley–Lieb relations).

[F2]

$τ_{n} (D)$ is $δ^{- n}$ times the evaluation of the closed matching graph obtained by joining corresponding exterior endpoints (Normalized Temperley–Lieb trace from diagram closure).

Proof

technique · Count the closed components in the glued matching graph

1.1F1F2givenalgebra

For basis diagrams $D, E : n \to n$ , the closed graph used to evaluate $τ_{n} (D E)$ is obtained by placing the two layers in a cyclic order and joining the exterior endpoints. Cutting this cyclic graph between the $D$ and $E$ layers gives the graph for $τ_{n} (E D)$ with the same components and the same number of closed loops. The normalizing power $δ^{- n}$ is the same, so $τ_{n} (D E) = τ_{n} (E D)$ . Bilinearity extends the identity to all $D, E$ .

2.1F1F2step 1.1algebra

In the closure of $x \otimes 1$ , the added vertical strand closes to one additional circle and is disjoint from the closed graph for $x$ . Thus its loop count is $c (x) + 1$ , while the normalizing exponent changes from $- n$ to $- (n + 1)$ ; the two powers of $δ$ cancel and $τ_{n + 1} (x \otimes 1) = τ_{n} (x)$ .

3.1F1F2step 2.1algebra∎

In the closure of $(x \otimes 1) d_{n}$ , the last cap and cup connect the new boundary strand to the preceding strand and create no additional closed component. The normalization still gains one factor $δ^{- 1}$ from rank $n$ to rank $n + 1$ . Hence $τ_{n + 1} ((x \otimes 1) d_{n}) = δ^{- 1} τ_{n} (x)$ on basis diagrams, and bilinearity gives it for every $x$ .

Depends on

Used by

Dependency tree · two levels

14 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

Joshua Chen, The Temperley-Lieb Categories and Skein Modules (standard reference, not scraped)
K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras (standard reference, not scraped)