Root-of-unity skein TQFT for 3-manifolds

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

$\zeta =A^2,q=-\zeta ,r=p\mathrm{/}2,\delta =-A^2-A^{-2}=q+q^{-1}\mathrm{.}$

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

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

The BHMV simple-label convention is the finite set

${\mathcal{J}}_p=\{0,1,\dots ,r-2\}\mathrm{.}$

The pivotal loop and coefficient conventions throughout this page are the displayed $\delta$ and $\mathbb{C}$; no zero quantum integer is inverted.

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

Let $p\ge 4$ be even, let $A$ be a primitive $2p$-th root of unity, and use the quantum integers and Temperley–Lieb loop value $\delta =-A^2-A^{-2}$ specified above. Write $r=p\mathrm{/}2$. The standard Jones–Wenzl recursion defines nonzero idempotents $p_n$ for $0\le n\le r-1$; their unnormalized closure traces are $[n+1]$. In particular, $p_{r-1}$ has trace zero and is negligible, while the next standard recursive formula for $p_r$ is undefined because its denominator is $[r]=0$. This statement concerns that recursion and does not assert that no differently constructed normalized idempotent can exist at rank $r$.

Root, loop, and quantum-integer conventions

Jones–Wenzl recursion and its exact denominator condition

Idempotence, corner, and closure-trace properties

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapter 3

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

Chen, The Temperley-Lieb Categories and Skein Modules, §5.3

Fix an even integer $p\ge 4$, a primitive $2p$-th root $A\in \mathbb{C}$, $r=p\mathrm{/}2$, and loop value $\delta =-A^2-A^{-2}$. Let ${\mathcal{T}}_{\delta }$ be the complex Temperley–Lieb diagram category with objects the nonnegative integers and morphisms finite linear combinations of perfect matchings of bottom and top boundary points with no intersecting arcs. Compose by stacking and remove each closed component with factor $\delta$. Define ${\mathcal{N}}_{\delta }(m,n)$ to be the subspace of morphisms $f:m\to n$ such that ${\mathrm{Tr}}_m(g\circ f)=0$ for every reverse morphism $g:n\to m$, where $\mathrm{Tr}$ is the unnormalized diagram-closure trace. Form the quotient category with the same objects and morphism spaces ${\mathrm{Hom}}_{{\mathcal{T}}_{\delta }}(m,n)\mathrm{/}{\mathcal{N}}_{\delta }(m,n)$. Its additive envelope adjoins finite formal direct sums, with matrix morphisms and matrix composition. Its idempotent completion adjoins objects $(X,e)$ for idempotents $e\in \mathrm{End}(X)$ with $e^2=e$, and morphisms $(X,e)\to (Y,f)$ are morphisms $a:X\to Y$ satisfying $a=fae$. Juxtaposition, planar cup-cap duality, and the induced trace descend to this completed quotient. This definition makes no assertion that it is semisimple, that its simple objects have a specified label set, or that any braid, twist, or modularity identities hold.

Root, loop, and quantum-integer conventions

Noncrossing matching diagrams and stacking

The exact closure-trace test defining negligible morphisms

Closure of negligible morphisms under composition and tensor product

Category and monoidal laws for the diagram operations

Fix even $p\ge 4$, a primitive $2p$-th root $A$, and $r=p\mathrm{/}2$. In the Temperley–Lieb diagram category over $\mathbb{C}$ with loop value $\delta =-A^2-A^{-2}$, call $f:m\to n$ negligible when ${\mathrm{Tr}}_m(g\circ f)=0$ for every reverse morphism $g:n\to m$, where $\mathrm{Tr}$ is the unnormalized diagram-closure trace. A tensor ideal is a family of linear subspaces of the morphism spaces closed under composition on either side and tensor product on either side by arbitrary morphisms. The negligible morphisms form a tensor ideal. This ideal is generated, as a tensor ideal, by the Jones–Wenzl projector $p_{r-1}$, defined by the standard recursion through its first zero quantum integer; in particular the quotient has precisely the surviving projector labels $0,\dots ,r-2$.

Root and loop conventions

Existence and zero trace of the boundary projector

Temperley–Lieb diagram category

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

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

The canonical-root quotient and its boundary ideal use the full-rank path decomposition and the exact diagram-to-tilting comparison. Both supplier statements and their complete local proofs were inspected. The proof is transported to other primitive roots by the finite cyclotomic-field argument in step 3.1; no source-only acceptance is asserted.

Full-rank path decomposition and root quotient

Exact diagram-to-tilting comparison

Let $p\ge 4$ be even, $A\in \mathbb{C}$ a primitive $2p$-th root, and $r=p\mathrm{/}2$. In the BHMV Temperley–Lieb quotient with loop value $\delta =-A^2-A^{-2}$ and planar pivotal trace, the complete simple-label set is $\{0,1,\dots ,r-2\}$, represented by the surviving Jones–Wenzl projectors $p_a$ obtained from the standard recursion using unit denominators $[2],\dots ,[a]$. Every simple is self-dual. Its fusion multiplicities are

$N_{ab}^c=1⟺\mathrm{\mid }a-b\mathrm{\mid }\le c\le \min (a+b,2r-4-a-b)\text{ and }a+b+c\text{ is even},$

and are zero otherwise. With the BHMV overcrossing map $c_i=A{1}_i+A^{-1}{U}_i$, where $U_i$ is the cup-cap generator, the full-framing ribbon structure has planar quantum dimensions $d_a=[a+1]$ and simple-object twists ${\theta }_a=(-1{)}^a{A}^{a(a+2)}$. Here the quantum integers use the fixed BHMV recurrence. The unnormalized Hopf-link matrix, whose $(a,b)$ entry is the pivotal trace of the double braiding on $p_a\otimes p_b$, is nondegenerate. Here modular means that this finite semisimple ribbon category has an invertible unnormalized Hopf-link matrix.

The quotient and planar pivotal convention

Exact root negligible ideal and simple labels

Root parameter, ribbon comparison, and parity correction

Positive-pivotal root fusion rule and self-duality

Positive-pivotal rank-one root category and sine matrix

Definition of modularity by the Hopf-link matrix

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§3–4

Gelca, Topological Quantum Field Theory with Corners Based on the Kauffman Bracket, §§3–4

Sawin, Quantum Groups at Roots of Unity and Modularity, §§1–5

For the fixed root parameter, let ${\mathcal{C}}_p$ denote the semisimplified Temperley–Lieb category with labels $0,\dots ,r-2$, pivotal dimensions $d_j=[j+1]$, and Hopf-link data inherited from the Kauffman bracket. Put ${\mathcal{D}}_p^2={\sum }_{j=0}^{r-2}{d}_j^2$ and choose ${\mathcal{D}}_p\in \mathbb{C}$ with that square. Choose $\kappa \in \mathbb{C}$ satisfying ${\kappa }^6=A^{-6-(p+1)p\mathrm{/}2}$. The root-specific extended data are the weighted Lagrangian ribbon bordisms whose objects are closed oriented labeled surfaces with a Lagrangian subspace of real first homology and whose morphisms are collared oriented labeled ribbon bordisms with integer weights. If $M_1:{\mathrm{\Sigma }}_0\to {\mathrm{\Sigma }}_1$ and $M_2:{\mathrm{\Sigma }}_1\to {\mathrm{\Sigma }}_2$ have relations ${\mathrm{\Lambda }}_{M_1},{\mathrm{\Lambda }}_{M_2}$ and boundary Lagrangians ${\lambda }_0,{\lambda }_1,{\lambda }_2$, define

$K_1={\mathrm{\Lambda }}_{M_1}({\lambda }_0),K_0={\lambda }_1,K_2={\mathrm{\Lambda }}_{M_2}^t({\lambda }_2),n(M_2\circ M_1)=n_2+n_1+\mu (K_1,K_0,K_2),$

using the Wall-signature Maslov convention. The category composition, units, and anomaly weight are those specified by this formula and the weighted Lagrangian bordism definition; no ordinary oriented anomaly-free functor is part of this Definition.

Root and loop conventions

Root category

Fusion, ribbon, and Hopf-link data

Weighted Lagrangian ribbon-bordism category and composition convention

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§1–2 and Appendix B

Pohorence, 3-manifold Invariants and TQFTs, §3.1

Walker, On Witten's 3-Manifold Invariants, §§1 and 18

Full choice assumption inherited by the weighted category

The BHMV source uses $p_1$-structures, while the registered prerequisite map specifies Lagrangian-weighted data. This Definition supplies the latter using the existing local category carrier; the equality between its Maslov correction and the BHMV $p_1$/signature anomaly remains a theorem obligation and is not asserted here.

Fix even $p\ge 4$, a primitive $2p$-th root $A$, the BHMV semisimplified category, and its quantum dimensions $d_j$ and twists ${\theta }_j$ for $0\le j\le r-2$. Put ${\mathrm{\Omega }}_p={\sum }_{j=0}^{r-2}{d}_j{p}_j$, ${\mathrm{\Delta }}_{+}={\sum }_j{d}_j^2{\theta }_j$, and ${\mathrm{\Delta }}_{-}={\sum }_j{d}_j^2{\theta }_j^{-1}$. The Statement asserts that both ${\mathrm{\Delta }}_{+}$ and ${\mathrm{\Delta }}_{-}$ are nonzero. For a finite integral framed link $L$ in the oriented three-sphere, color each surgery component by ${\mathrm{\Omega }}_p$ and normalize the bracket evaluation by ${\mathrm{\Delta }}_{+}^{b_{+}(L)}{\mathrm{\Delta }}_{-}^{b_{-}(L)}$, where $b_{\pm }(L)$ are the positive and negative indices of the framing matrix. With the stated BHMV root and weighted anomaly convention, the resulting value is unchanged under framed isotopy, handle slides, and addition or deletion of a split $\pm 1$-framed unknot.

Root, coefficient, and loop conventions

Simple projectors and category convention

Fusion dimensions and twists

Weighted anomaly normalization

Blanchet, Habegger, Masbaum, and Vogel, Three-Manifold Invariants Derived from the Kauffman Bracket, §§2–4

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 8 and 12

Pohorence, 3-manifold Invariants and TQFTs, §3.1

Assume full AC for the surgery-presentation and universal-construction inputs below. Fix an even integer $p\ge 4$, a primitive $2p$-th root $A\in \mathbb{C}$, put $r=p\mathrm{/}2$ and $\delta =-A^2-A^{-2}$, and use the selected BHMV sixth root $\kappa$ satisfying ${\kappa }^6=A^{-6-(p+1)p\mathrm{/}2}$. Fix a closed oriented marked surface $\mathrm{\Sigma }$ with its Lagrangian and weighted anomaly data. Let $I_{\mathrm{\Sigma }}$ be the finite union of its marked ribbon-end bands, with each band labeled by its specified simple object $a_j\in \{0,\dots ,r-2\}$. For connected $\mathrm{\Sigma }$, fix marked handlebodies $H,H^{\mathrm{\prime }}$ whose boundaries are both identified orientation-preservingly with $\mathrm{\Sigma }$, so that $H{\cup }_{\mathrm{\Sigma }}(-H^{\mathrm{\prime }})$ is the standard oriented $S^3$. Both filling modules carry the specified boundary labels before reversal; orientation reversal of the second filling dualizes its boundary labels and reverses its ribbon-end arcs. Disconnected and empty surfaces are included by the tensor and unit convention below.

For a banded boundary link $I$ in $\mathrm{\partial }H$, let ${\mathcal{K}}_A(H;I)$ be the complex specialization of the relative Jones–Kauffman skein module generated by framed banded links in $H$ meeting $\mathrm{\partial }H$ transversely in exactly $I$, modulo framed isotopy fixed on $\mathrm{\partial }H$ and the following local relations in interior balls. In a chart where the over-strand runs southwest to northeast and the under-strand northwest to southeast, let $L_A$ join the two left endpoints and the two right endpoints, and let $L_B$ join the top endpoints and the bottom endpoints, and require the crossing and both smoothings to carry untwisted blackboard framing, with no additional full twists and matching framed endpoint collars. Impose $[L_{\times }]=A[L_A]+A^{-1}[L_B]$. If $O$ is a zero-framed trivial circle bounding a disk disjoint from the rest of the link, impose $[L\bigsqcup O]=\delta [L]$. The empty link is allowed.

Let ${\tilde{I}}_{\mathrm{\Sigma }}$ be the boundary link obtained by replacing each band $I_j$ by $a_j$ parallel bands, deleting it when $a_j=0$. On ${\mathcal{K}}_A(H;{\tilde{I}}_{\mathrm{\Sigma }})$, the identity-coefficient Jones–Wenzl projector $p_{a_j}$ acts on the corresponding parallel strands in a boundary collar. Define the colored relative skein module by ${\mathcal{K}}_A(H;I_{\mathrm{\Sigma }},\mathbf{a})=P_{\mathbf{a}}{\mathcal{K}}_A(H;{\tilde{I}}_{\mathrm{\Sigma }})$, where $P_{\mathbf{a}}$ is the product of these mutually disjoint boundary projectors. Every denominator in these projectors is among $[2],\dots ,[a_j]$, which are nonzero because $a_j\le r-2$. When $I_{\mathrm{\Sigma }}$ is empty, this is the ordinary closed-link module ${\mathcal{K}}_A(H)$.

Let $\mathfrak{r}:{\mathcal{K}}_A(H^{\mathrm{\prime }};I_{\mathrm{\Sigma }},\mathbf{a})\to {\mathcal{K}}_A(-H^{\mathrm{\prime }};-I_{\mathrm{\Sigma }},{\mathbf{a}}^{\ast })$ be the semilinear orientation-reversal map: for a finite skein sum, $\mathfrak{r}({\sum }_L{c}_L[L])={\sum }_L\overline{c_L}[-L]$, with the second filling, its ribbon data and anomaly data reversed. The bar is complex conjugation, so $\bar{A}=A^{-1}$ and $\bar{\kappa }={\kappa }^{-1}$. Reversal exchanges the two bracket smoothings, and conjugation exchanges their coefficients, so $\mathfrak{r}$ respects the skein relations and the boundary projectors. For $x\in {\mathcal{K}}_A(H;I_{\mathrm{\Sigma }},\mathbf{a})$ and $y\in {\mathcal{K}}_A(H^{\mathrm{\prime }};I_{\mathrm{\Sigma }},\mathbf{a})$, glue $x$ to $\mathfrak{r}(y)$ along the matching boundary bands and idempotents. Denote the resulting graph by $\mathrm{\Gamma }(x,\mathfrak{r}(y))$. Define $B_{\mathrm{\Sigma }}(x,y)=Z_p(S^3,\mathrm{\Gamma }(x,\mathfrak{r}(y)),w)$ with the prescribed gluing weight. This is linear in $x$ and conjugate-linear in $y$, as in BHMV Proposition 1.9, p.891. Ordinary gluing of two already oppositely oriented skein inputs is instead bilinear; the reversal here is part of the Definition. Define

$\mathrm{Rad}(B_{\mathrm{\Sigma }})=\{x\in {\mathcal{K}}_A(H;I_{\mathrm{\Sigma }},\mathbf{a}):B_{\mathrm{\Sigma }}(x,y)=0\text{ for every }y\in {\mathcal{K}}_A(H^{\mathrm{\prime }};I_{\mathrm{\Sigma }},\mathbf{a})\},V_p(\mathrm{\Sigma })={\mathcal{K}}_A(H;I_{\mathrm{\Sigma }},\mathbf{a})\mathrm{/}\mathrm{Rad}(B_{\mathrm{\Sigma }})\mathrm{.}$

This is the radical-quotient handlebody state-space construction for a connected marked surface. For a finite disjoint union $\mathrm{\Sigma }={⨆}_{a=1}^k{\mathrm{\Sigma }}_a$ with the direct-sum component reference Lagrangian, define $V_p(\mathrm{\Sigma })={⨂}_{a=1}^k{V}_p({\mathrm{\Sigma }}_a)$, and define its pairing as the product of the component sesquilinear pairings on pure tensors. Set $V_p(\mathrm{\varnothing })=\mathbb{C}$ and $B_{\mathrm{\varnothing }}(z,w)=z\bar{w}$. Permuting components uses the ordinary tensor symmetry; the empty tensor is the unit. These are the external BHMV conventions (Q1) p.885 and (M) p.886, whose validity for even $p\ge 4$ is part of Theorem 1.4 p.887; Gilmer–Masbaum §11 p.24 also states the empty-surface unit. For an arbitrary specified real Lagrangian, including one that does not split over components, use the weight-zero identity cylinder from the componentwise handlebody-kernel reference to that Lagrangian in the registered weighted theory. The reverse cylinder is its inverse since $\mu (L,L^{\mathrm{\prime }},L)=0$, exactly as in the existing real-reference transport for weighted gluing; this does not assume every real Lagrangian splits or is rational. The tensor conventions are taken at the reference objects and then transported by these inverse cylinders. Independence of the marking, finite generation, nondegeneracy and the gluing maps remain separate mathematical obligations supplied below by the stated external BHMV/weighted theory inputs.

External normalization input. In this Definition the closed marked evaluation is the BHMV TQFT normalization, not merely the raw signature-normalized closed invariant ${\theta }_p$ of the Kirby lemma. BHMV §2 pp.897–898 and Gilmer–Masbaum §11 pp.23–25 give $Z_p(M,\mathrm{\Gamma },w)={\eta }^{b_0(M)+b_1(M)}({\kappa }^3{)}^w{\theta }_p(M,\mathrm{\Gamma }),{\mathrm{\Delta }}_{\pm }={\eta }^{-1}{\kappa }^{\pm 3}\mathrm{.}$ Let $g_p=\frac{1}{2}{\sum }_{m=1}^{2p}(-1{)}^m{A}^{m^2}$, and let the bar denote the BHMV involution $A\mapsto A^{-1}$, $\kappa \mapsto {\kappa }^{-1}$. The normalizing scalar is $\eta =(A\kappa {)}^3(A^2-A^{-2})p^{-1}\overline{g_p}\in {\mathbb{C}}^{\times }$. Thus $Z_p(S^3,\mathrm{\varnothing },0)=\eta$ and $Z_p(S^2\times S^1,\mathrm{\varnothing },0)=1$. These nonzero normalizations and the registered Wall-sign weight are part of the pairing. BHMV Proposition 1.9, p. 891, applies to a general prescribed banded boundary link $I$: the map from the relative skein module to the universal state space is surjective and its kernel is the left radical of the complementary pairing. For colored boundary bands, BHMV §§4.5–4.8 realize each color by the corresponding parallel cable and Jones–Wenzl projector. BHMV Theorem 4.11, p. 909, states that admissibly colored relative spines whose boundary is the prescribed colored link form an orthogonal basis. The selected $\kappa$ remains a sixth root, with one weight step acting by its cube. The source is used at these located statements and constructions, not as a claimed locally reconstructed universal-construction proof.

Source hypotheses for this identification. The normalized BHMV closed invariant is multiplicative and involutive, so BHMV Proposition 1.1 gives its cobordism-generated quantization functor at this fixed root; extend its coefficients to $\mathbb{C}$ as stated above. BHMV Theorem 1.7 supplies both hypotheses needed for Proposition 1.9: its Kauffman bracket relations let that proposition replace the free relative bordism module by the specialized relative skein module, and its surgery axioms give the surjectivity and radical-kernel description. BHMV's colored expansion in §§4.5–4.8 transports the same construction to the boundary projectors defining ${\mathcal{K}}_A(H;I_{\mathrm{\Sigma }},\mathbf{a})$. The corresponding weighted Lagrangian version is the BHMV skein TQFT with integral weights and Lagrangians in place of $p_1$-structures, as described by Gilmer–Masbaum in §11; their one-weight multiplier is matched to the selected BHMV sixth root by ${\kappa }_{\mathrm{G}\mathrm{M}}={\kappa }^3$. In that section Lemma 11.1 states the weighted surgery axiom and Remark 11.3 applies it to the handlebody skein presentation; the source refers the Lemma 11.1 proof to the BHMV argument rather than repeating it. These are imported source inputs, not a locally reconstructed proof, and they do not add another quotient or change the state-space interface.

Framed-link Kauffman bracket module

BHMV, relative Jones–Kauffman skein module and Proposition 1.9, §§1.C and 2, pp. 889–891

BHMV, colored expansion and colored-spine basis, §§4.5–4.8 and Theorem 4.11, pp. 905–909

Root and coefficient conventions

Weighted anomaly data

Closed surgery invariant required by the pairing

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§1–2

BHMV, Propositions 1.1 and 1.9 and Theorem 1.7, §§1.A, 1.C and pp. 886–891

Gilmer–Masbaum, Maslov Index, Lagrangians, Mapping Class Groups and TQFT, §11, weighted BHMV construction, Lemma 11.1 and Remark 11.3

Pohorence, 3-manifold Invariants and TQFTs, §§1.2 and 3.1–3.3

Chen, The Temperley-Lieb Categories and Skein Modules, Chapter 6

Let $(H,I_{\mathrm{\Sigma }},\mathbf{a})$ be a marked handlebody with prescribed boundary bands $I_{\mathrm{\Sigma }}$ and boundary labels $\mathbf{a}$. Fix a finite banded relative spine $G\subset H$, meaning a banded graph whose degree-one boundary vertices give exactly $I_{\mathrm{\Sigma }}$, whose interior vertices have degree two or three, and whose regular neighborhood is $H$; for the unmarked $3$-ball, the empty graph is allowed. Let each edge carry a label in $\{0,\dots ,r-2\}$, with every boundary half-edge on a band $I_j$ labeled by its prescribed color $a_j$. Call a labeling admissible when labels agree across every interior $2$-valent vertex and the three labels at each interior trivalent vertex satisfy the root fusion rule. In the radical-quotient state space $V_p(\mathrm{\Sigma })$, the states represented by admissibly colored relative spines span, and the set of such labelings is finite. This statement asserts spanning, not linear independence.

Handlebody state space

Root fusion rule

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§4–5

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 4–7

Chen, The Temperley-Lieb Categories and Skein Modules, Chapter 6

For the fixed root parameter, the normalized BHMV sesquilinear pairing between the relative skein modules of two complementary marked handlebodies induces a perfect pairing between their radical quotients, using the coefficient-conjugate space on the oppositely oriented side. The inverse pairing defines the sewing map for gluing along a closed marked surface, and these maps satisfy the stated weighted Lagrangian anomaly correction.

State-space pairing

Finite spine spanning set

Weighted Lagrangian anomaly convention

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§1 and 4–5

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 6–7

Pohorence, 3-manifold Invariants and TQFTs, §3

Assume full AC. Fix even $p\ge 4$, a primitive $2p$-th root $A$, and the BHMV Kauffman bracket convention with $\delta =-A^2-A^{-2}$. 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.

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