Kauffman bracket skein modules of small 3-manifolds

Abstract

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 $\mathbb{Q}(A)$.

44pages, 1 figure. Revisions following referee reports. To appear in Advances in Mathematics

References in corpus (10)

• Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories
• Skein modules of 3-manifolds
• Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations
• On Kauffman Bracket Skein Modules at Root of Unity
• Skein spaces and spin structures
• On representation varieties of 3-manifold groups
• The SL(2,C) Casson invariant for Seifert fibered homology spheres and surgeries on twist knots
• On the Kauffman bracket skein module of the quaternionic manifold
• Infinite families of hyperbolic $3$-manifolds with finite dimensional skein modules
• Non-semisimple quantum invariants and abelian classical shadows