The finiteness conjecture for skein modules

Abstract

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.

Gromov norm and Turaev-Viro invariants of 3-manifolds

“…By [AM20, Theorem 1.4], if X (M ) is finite and reduced, this later dimension is |X(M )|. Therefore, in this setting, Theorem 1.1 provides evidence for the conjecture in [GJS23].…”

“…However, a proof of Theorem 2.2 appears in [KK22, Proof of Theorem 6.3]. The proof relies on the fact that the Azumaya locus of the skein algebra of a closed surface was shown to contain all irreducible characters, [FKBL19,GJS23], and more recently also shown to contain all abelian non-central characters, [KK22]. A character is abelian (resp.…”

“…They received particular attention in the recent years [BW16, FKBL19, GJS19, GJS23], due to their connections to quantum groups, cluster algebras, quantum field theories, and other areas of mathematics and physics. Witten conjectured and Gunningham, Jordan, and Safronov proved [GJS23], that the skein module S(M ) is finite dimensional for any closed 3-manifold M . However, their work doesn't offer an effective method for computing the dimension dim Q(A) S(M ).…”

“…Conjecture 1.1 is inspired and closely related to a conjecture of Przytycki (Conjecture (E) of [Kir97, Problem 1.92]) asserting that S(M, Q[A ±1 ]) is free for manifolds M containing no essential surfaces. Since S(M ) := S(M, Q(A)) is finitely generated over Q(A) by [GJS23],…”

Langlands duality for skein modules of 3-manifolds

“…The grading and twisting constructions are dual in a precise sense, which we will soon see. For now let us outline the construction of twisting of skein modules: the complete details are presented in the forthcoming [GJS23], so we will be somewhat brief. Definition 3.1.…”