Classification of tight contact structures on surgeries on the figure-eight knot

Abstract

Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, whether we can classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic $3$–manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.