Tilting modules arising from knot invariants

Abstract

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 = ⊕ i M ( 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 a 1 , a 2 . We show that the module M is rigid and τ -rigid, and we construct a completion of M to a tilting (and τ -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 , q , r (a tree with three branches). This quiver is of finite type if ( a 1 , a 2 ) = ( a 1 , 2 ) , ( 1 , a 2 ) , or ( 2 , 3 ) , it is tame for ( a 1 , a 2 ) = ( 2 , 4 ) or ( 3 , 3 ) , and wild otherwise.