• DOI:10.4310/CAG.2016.V24.N3.A1
• Corpus ID: 74871907

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

@article{Baker2013SomeKI,
  title={Some knots in S^1 x S^2 with lens space surgeries},
  author={Kenneth L. Baker and Dorothy Buck and Ana G. Lecuona},
  journal={arXiv: Geometric Topology},
  year={2013}
}

• K. Baker, D. Buck, Ana G. Lecuona
• Published 27 February 2013
• Mathematics
• arXiv: Geometric Topology

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.