Tilting modules arising from knot invariants

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