Skein TQFT state space and handlebody pairing

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