Free genus one knots with large volume

Abstract

A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genus one. This implies that there are knots with free genus one but arbitrarily large canonical genus, and that there exist knots admitting incompressible free Seifert surfaces which cannot be built by applying Seifert's algorithm to a projection of the knot.