Families of knots which produce diffeomorphic Fintushel-Stern surgeries

Fintushel-Stern's knot surgery construction on elliptic surfaces has been a central source of exotic, smooth four-manifolds since its introduction in the 1990's. The construction associates a homotopy elliptic surface to a classical knot. These homotopy elliptic surfaces are non-diffeomorphic if the knots used in their constructions have different Alexander polynomials, and the converse has been an open question for twenty years. This talk will give some results which represent partial progress in favor of this question.