We propose a classification of knots in S1 x S2 that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knots in S1 x S2 may be obtained from a Berge-Gabai knot in a Heegaard solid torus of S1 x S2, 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 S1 x S2 are both doubly primitive and spherical braids. This classification arose from generalizing Berge's list of doubly primitive knots in S3, 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 S1 x S2 and reduce this conjecture to classifying the homology classes of knots in lens spaces admitting a longitudinal S1 x S2 surgery.
|Number of pages||35|
|Publication status||Published - 27 Feb 2013|