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Root-of-unity skein TQFT gluing and surgery invariance

Statement

Assume full AC. Fix even $p \geq 4$ , a primitive $2 p$ -th root $A$ , and the BHMV Kauffman bracket convention with $δ = - A^{2} - A^{- 2}$ . On the weighted Lagrangian extended bordism category built from the root semisimplified Temperley–Lieb data, the normalized Kirby-color surgery maps define a finite-dimensional skein TQFT: the handlebody state spaces are the radical quotients of their skein modules, gluing is given by the inverse normalized handlebody pairing, and the resulting closed value is unchanged under isotopy, framed handle slides, and split $\pm 1$ -blow-ups. Forgetting the weight and Lagrangian extension yields only the projective ordinary-oriented data unless a separate anomaly-cancellation result is proved.

Statement references

Root and bracket conventions

Weighted Lagrangian extension and anomaly data

State spaces

Kirby-color surgery maps

Pairing and gluing identities

Framed surgery presentations and move classification

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, Theorem 1.4 and §§1–5

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 7 and 12

Pohorence, 3-manifold Invariants and TQFTs, §§1.2 and 3

Facts & Assumptions

Given: Full AC, the fixed BHMV root and pivotal convention, weighted Lagrangian bordism data, the root state spaces and their surgery maps.

[A1]

The Axiom of Choice supplies the full choice assumption used by the surgery-presentation and Kirby-move supplier.

[F1]

Kirby moves classify integral framed-link presentations: framed surgery diagrams for the same closed oriented manifold are connected by the stated isotopies, framed handle slides, and split signed-unit-unknot moves.

[F2]

Root Kirby-color handle-slide and blow-up invariance supplies the closed surgery invariance, nonzero Gauss sums and precise external BHMV weighted normalization, including weight unit $κ^{3}$ .

[F3]

Handlebody pairing nondegeneracy and gluing supplies the perfect radical-quotient pairing, finite inverse sewing tensor and externally specified weighted gluing.

[F4]

Weighted Lagrangian ribbon bordisms supplies associative weighted Lagrangian composition with the Wall-signature Maslov correction.

Proof

Given: The data in the Statement.

1.1A1F2F3F4givenconstruct

For a fixed surgery presentation, the color set is finite and the state-space pairing is defined by a finite bracket evaluation. This constructs a candidate linear map for each presented weighted bordism.

2.1A1F1F2F3F4step 1.1

Any two presented closed surgery links are related by [F1], and [F2] preserves their values under every move. For bordisms, use the BHMV additional $η^{b_{0} + b_{1}}$ normalization specified in the state-space Definition, not the raw closed invariant alone. The finite state spaces, perfect pairing and actual surgery/sewing maps are [F3], whose external BHMV/Gilmer–Masbaum input supplies the TQFT construction at the exact even root and complex specialization. The inverse-pairing contraction composes these maps, with the integer Wall/Maslov weight law [F4]. This proves finite-dimensionality, gluing and presentation invariance of the stated weighted theory.

3.1A1F2F3F4step 2.1∎

Forgetting the weight removes the scalar that corrects the Maslov/signature defect. Therefore the imported construction descends only projectively to ordinary oriented data without a further anomaly-cancellation input. No such additional theorem is asserted here. Full AC remains the declared branch of the presentation/marked category interfaces; finite color sums add no stronger assumption.

Depends on

Used by

Worked root-of-unity fusion and torus-state cases Example

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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), Theorem 1.4 and §§1–5 (standard reference, not scraped)
Louis H. Kauffman and Sostenes L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 7 and 12 (standard reference, not scraped)
Sean Pohorence, 3-manifold Invariants and TQFTs (January 2016), §§1.2 and 3 (standard reference, not scraped)
BHMV, Topological Quantum Field Theories Derived from the Kauffman Bracket, Topology 34 (1995), exact external inputs (standard reference, not scraped)
Gilmer–Masbaum, Maslov index, Lagrangians, Mapping Class Groups and TQFT, arXiv:0912.4706v3 (standard reference, not scraped)