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Kauffman bracket skein modules of small 3-manifolds
The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $$3$$-manifolds are finitely generated over $$\Q(A)$$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $$S(M,\Q[A^\pmo])$$ of $$M$$ is tame (e.g. finitely generated over $$\Q[A^{\pm 1}]$$), and the $$SL(2, \C)$$-character scheme is reduced, then the dimension $$\dim_{\Q(A)}\, S(M, \Q(A))$$ is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating $$\dim_{\Q(A)}\, S(M, \Q(A))$$ to the Abouzaid-Manolescu $$SL(2,\C)$$-Floer theoretic invariants, for infinite families of 3-manifolds. We prove a criterion for reducedness 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 $$\Q(A)$$.
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- Award ID(s):
- 2304033
- PAR ID:
- 10631577
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 467
- Issue:
- C
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 110169
- Subject(s) / Keyword(s):
- Skein moduleCharacter variety3-manifoldKauffman bracket primary57K31secondary57K16
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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