Heegaard splittings of knot and link complements

Abstract

We mainly focus on the stabilization problem of Heegaard splittings of knot and link complements in $S^3$. We give a condition for a pair of unknotting tunnels of a non-trivial tunnel number one link to give a genus three Heegaard splittings of the link complement and show that every 2-bridge link has such a pair of unknotting tunnels. For the tunnel number one knot, we give a more restrictive condition for a pair of unknotting tunnels of a non-trivial tunnel number one knot to give a genus three Heegaard splittings of the knot complements and show that every 2-bridge knot has such a pair of unknotting tunnels. In addition, we consider the disjoint curve property for Heegaard splittings of tunnel number one knot complements. We also list the knot types or types of core of the compression body having the d.c.p(disjoint curve property).