Knots with distinct primitive/primitive and primitive/Seifert representatives

TL;DR

This paper constructs families of knots with distinct primitive/Seifert and primitive/primitive embeddings in a genus 2 Heegaard surface, sharing the same surface slope, expanding understanding of knot types related to lens spaces and Seifert fibered spaces.

Contribution

It introduces new two-parameter families of knots with different primitive/Seifert and primitive/primitive embeddings at the same surface slope, highlighting their coexistence.

Findings

Constructed a two-parameter family of knots with distinct primitive/Seifert embeddings.

Identified torus knots with both primitive/primitive and primitive/Seifert representatives at the same slope.

Abstract

Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, $F$, in $S^3$; 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.