Tilting modules arising from knot invariants

We construct tilting modules over Jacobian algebras arising from knots. To a two bridge knot L[a(1), ... , a(n)], we associate a quiver Q with potential and its Jacobian algebra A. We construct a family of canonical indecomposable A-modules M(i), each supported on a different specific subquiver Q(i) of Q. Each of the M(i) is expected to parametrize the Jones polynomial of the knot. We study the direct sum M = circle plus M-i(i) of these indecomposables inside the module category of A as well as in the cluster category. In this paper we consider the special case where the two-bridge knot is given by two parameters a1, a2. We show that the module M is rigid and & UTau;-rigid, and we construct a completion of M to a tilting (and tau-tilting) A-module T. We show that the endomorphism algebra End(A) T of T is isomorphic to A, and that the mapping T ? A[1] induces a cluster automorphism of the cluster algebra A(Q). This automorphism is of order two. Moreover, we give a mutation sequence that realizes the cluster automorphism. In particular, we show that the quiver Q is mutation equivalent to an acyclic quiver of type T-p,T-q,T-r (a tree with three branches). This quiver is of finite type if (a(1), a2) = (a(1), 2), (1, a2), or (2, 3), it is tame for (a(1), a2) = (2, 4) or (3, 3), and wild otherwise. (C) 2022 Elsevier B.V. All rights reserved.