Kauffman bracket skein modules of small 3-manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908344834326528 |
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| author | Detcherry, Renaud Kalfagianni, Efstratia Sikora, Adam S. |
| author_facet | Detcherry, Renaud Kalfagianni, Efstratia Sikora, Adam S. |
| contents | The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\mathbb Q(A)$. In this paper, we develop a novel method for computing these skein modules.
We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame (e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb
C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large families of 3-manifolds.
We also prove a criterion for reduceness of character varieties of closed $3$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds.
We also prove that the skein modules of rational homology spheres have dimension at least $1$ over $\mathbb Q(A)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_16188 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Kauffman bracket skein modules of small 3-manifolds Detcherry, Renaud Kalfagianni, Efstratia Sikora, Adam S. Geometric Topology Quantum Algebra Representation Theory 57K31, 57K16 The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\mathbb Q(A)$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame (e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large families of 3-manifolds. We also prove a criterion for reduceness of character varieties of closed $3$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least $1$ over $\mathbb Q(A)$. |
| title | Kauffman bracket skein modules of small 3-manifolds |
| topic | Geometric Topology Quantum Algebra Representation Theory 57K31, 57K16 |
| url | https://arxiv.org/abs/2305.16188 |