Geometry of (1,1)-Knots and Knot Floer Homology

Abstract

We apply the technique of Heegaard Floer Homology to (1,1)-knots (1-bridge knots on the torus) to determine all (1,1)-knots of crossing number up to 12. We also prove miscellaneous results regarding (1,1)-knots, including the existence of a family with trivial Alexander polynomial, and symmetry results. In addition, we define and study a new class of multi-pointed Heegaard diagrams for links in S^3 that generalizes the classical notion of grid diagrams.