Some knots in S^1 x S^2 with lens space surgeries

TL;DR

This paper classifies certain knots in S^1 x S^2 that admit lens space surgeries, identifying new families beyond known Berge-Gabai knots, and conjectures a complete list of such doubly primitive knots.

Contribution

It introduces two new families of knots in S^1 x S^2 with lens space surgeries and extends the classification of doubly primitive knots beyond previous work.

Findings

Identified two new families of knots with lens space surgeries in S^1 x S^2.

All classified knots are doubly primitive and spherical braids.

Conjectured a complete classification of such knots in S^1 x S^2.

Abstract

We propose a classification of knots in S^1 x S^2 that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knots in S^1 x S^2 may be obtained from a Berge-Gabai knot in a Heegaard solid torus of S^1 x S^2, as observed by Rasmussen. We show that there are yet two other families of knots: those that lie on the fiber of a genus one fibered knot and the `sporadic' knots. All these knots in S^1 x S^2 are both doubly primitive and spherical braids. This classification arose from generalizing Berge's list of doubly primitive knots in S^3, though we also examine how one might develop it using Lisca's embeddings of the intersection lattices of rational homology balls bounded by lens spaces as a guide. We conjecture that our knots constitute a complete list of doubly primitive knots in S^1 x S^2 and reduce this conjecture to classifying the homology classes of knots in lens spaces admitting a longitudinal S^1 x S^2 surgery.