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The BHMV root-of-unity skein parameter

Definition

Fix an even integer $p \geq 4$ and a primitive $2 p$ -th root of unity $A \in C$ . Put

$ζ = A^{2}, q = - ζ, r = p / 2, δ = - A^{2} - A^{- 2} = q + q^{- 1} .$

The coefficient field after specialization is $C$ , and the Kauffman bracket loop coefficient is $δ$ in the framed-link relation $[L ⊔ O] = δ [L]$ . Define the quantum integers by $[0] = 0$ , $[1] = 1$ , and $[j + 1] = δ [j] - [j - 1]$ for $j \geq 1$ . Since $q^{2} = A^{4}$ has order $r \geq 2$ , these quantum integers also satisfy

$[j] = \frac{q^{j} - q^{- j}}{q - q^{- 1}} = (- 1)^{j - 1} \frac{A^{2 j} - A^{- 2 j}}{A^{2} - A^{- 2}} .$

The BHMV simple-label convention is the finite set

$J_{p} = {0, 1, \dots, r - 2} .$

The pivotal loop and coefficient conventions throughout this page are the displayed $δ$ and $C$ ; no zero quantum integer is inverted.

Definition references

Framed-link skein module and its loop coefficient

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§1–3

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 2–3

Chen, The Temperley-Lieb Categories and Skein Modules, §§2.1 and 5.1–5.4

Depends on

Kauffman bracket skein module of a three-manifold

Used by

The next Jones–Wenzl recursion denominator vanishes at p=6 Counterexample
Semisimplified Temperley–Lieb quotient at the BHMV root Definition
Skein TQFT state space and handlebody pairing Definition
Weighted Lagrangian extension and root anomaly data Definition
Worked root-of-unity fusion and torus-state cases Example
Jones–Wenzl projectors exist up to the first zero quantum integer Lemma
Root fusion rules and completeness of the simple objects Lemma
Root Jones–Wenzl recursion and its negligible ideal Lemma
Root Kirby-color handle-slide and blow-up invariance Lemma

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Sources

C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, Topology 34 (1995), §§1–3 (standard reference, not scraped)
Louis H. Kauffman and Sostenes L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 2–3 (standard reference, not scraped)
Joshua Chen, The Temperley-Lieb Categories and Skein Modules, §§2.1 and 5.1–5.4 (standard reference, not scraped)