ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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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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At loop value zero the normalized closure trace is undefined

Example

Set $R = C$ and $δ = 0$ . The unnormalized planar closure ${Tr}_{n}$ remains defined, but no normalized closure trace $τ_{n} = δ^{- n} {Tr}_{n}$ exists. Already at $n = 1$ , the identity has closure value ${Tr}_{1} (1_{1}) = 0$ , so a scalar normalization of this closure cannot satisfy $τ_{1} (1_{1}) = 1$ .

Facts & Assumptions

Given: The loop value $δ = 0$ and the one-strand identity diagram.

[F1]

The unnormalized closure assigns $δ^{c (D)}$ to each closed diagram and requires no unit hypothesis; normalization is defined only when $δ$ is a unit (Unnormalized diagram closure and partial trace).

[F2]

The closure of $1_{1}$ has one circle, so ${Tr}_{1} (1_{1}) = δ$ (Diagram closure trace is cyclic and satisfies the tower rules, Temperley–Lieb planar diagram data and endomorphism modules).

Verification

technique · direct calculation

1.1F1F2given

At $n = 1$ , closing the single vertical strand joins its top and bottom endpoints to make exactly one circle. By [F1] and [F2], ${Tr}_{1} (1_{1}) = δ^{1} = 0$ .

2.1F1step 1.1algebra∎

Any normalized trace must send the algebra identity to $1$ , but the proposed scalar multiple $δ^{- 1} {Tr}_{1}$ is undefined because $0$ has no multiplicative inverse in $C$ . Even replacing it by any scalar multiple $c {Tr}_{1}$ gives $c {Tr}_{1} (1_{1}) = c \cdot 0 = 0$ , never $1$ . Hence the unnormalized closure exists while the normalized closure does not.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)