Knots with distinct primitive/primitive and primitive/Seifert representatives

Details Knots with distinct primitive/primitive and primitive/Seifert representatives Knots with distinct primitive/primitive and primitive/Seifert representatives

2009-09-02

arxiv.org/abs/0909.0476v4

Mathematics

Geometric Topology

12 pages, 8 figures; 2 figures updated

Scientific paper

10.1142/S0218216511009625

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.

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