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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.19352 |
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| _version_ | 1866917117367943168 |
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| author | Goertz, Leon J. |
| author_facet | Goertz, Leon J. |
| contents | For a commutative Frobenius algebra $A$, we construct a $(2,3,3+\varepsilon)$-dimensional TQFT $\mathsf{AFK}_A$ that assigns to a 3-manifold a skein module of embedded $A$-decorated surfaces. These surface skein modules have been first defined by Asaeda--Frohman and Kaiser using skein relations that generalize the combinatorics of Bar-Natan's dotted cobordisms. For 3-manifolds with boundary, we show that surface skein modules carry an action by a certain surface skein category associated with the boundary, which yields a gluing formalism. Our main result concerns a partial extension of $\mathsf{AFK}_A$ to dimension 4, which uses an inductive state-sum construction following Walker. As an example, the equivariant version of Lee's deformation of dotted cobordisms yields a TQFT that extends to 4-dimensional 2-handlebodies but not 3-handlebodies. Finally, we characterize the attachment of 4-dimensional 2-handles by means of a certain Kirby color and use it to compute the invariants of 4-dimensional 2-handlebodies in examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies Goertz, Leon J. Quantum Algebra Geometric Topology For a commutative Frobenius algebra $A$, we construct a $(2,3,3+\varepsilon)$-dimensional TQFT $\mathsf{AFK}_A$ that assigns to a 3-manifold a skein module of embedded $A$-decorated surfaces. These surface skein modules have been first defined by Asaeda--Frohman and Kaiser using skein relations that generalize the combinatorics of Bar-Natan's dotted cobordisms. For 3-manifolds with boundary, we show that surface skein modules carry an action by a certain surface skein category associated with the boundary, which yields a gluing formalism. Our main result concerns a partial extension of $\mathsf{AFK}_A$ to dimension 4, which uses an inductive state-sum construction following Walker. As an example, the equivariant version of Lee's deformation of dotted cobordisms yields a TQFT that extends to 4-dimensional 2-handlebodies but not 3-handlebodies. Finally, we characterize the attachment of 4-dimensional 2-handles by means of a certain Kirby color and use it to compute the invariants of 4-dimensional 2-handlebodies in examples. |
| title | A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies |
| topic | Quantum Algebra Geometric Topology |
| url | https://arxiv.org/abs/2511.19352 |