KNOTS WITH DISTINCT PRIMITIVE/PRIMITIVE AND PRIMITIVE/SEIFERT REPRESENTATIVES

Abstract

Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, F, in S3; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. Here we construct a two-parameter family of knots that have distinct primitive/Seifert embeddings in F with the same surface slope, as well as a family of torus knots that have a primitive/primitive representative and a primitive/Seifert representative with the same surface slope.