Seifert fibered surgeries with distinct primitive/Seifert positions

TL;DR

This paper constructs new infinite families of Seifert fibered surgeries on hyperbolic knots with arbitrarily large volume, each having distinct primitive/Seifert positions, expanding understanding of knot surgeries in 3-sphere.

Contribution

It introduces novel infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions, including hyperbolic knots with large volume, from a new perspective.

Findings

Constructed infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions.

Included hyperbolic knots with arbitrarily large volume in these families.

Extended the classification of Seifert fibered surgeries beyond known examples.

Abstract

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view. In particular, we can choose such Seifert surgeries (K, m) so that K is a hyperbolic knot whose complement S^3 - K has an arbitrarily large volume.