Communications in Analysis and Geometry

Volume 24 (2016)

Number 3

Some knots in $S^1 \times S^2$ with lens space surgeries

Pages: 431 – 470



Kenneth L. Baker (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Dorothy Buck (Department of Mathematics, Imperial College London, South Kensington, London, United Kingdom)

Ana G. Lecuona (Aix Marseille Université, CNRS, Marseille, France)


We propose a classification of knots in $S^1 \times S^2$ that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knot in $S^1 \times S^2$ may be obtained from a Berge–Gabai knot in a Heegaard solid torus of $S^1 \times S^2$, 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. Assuming results of Cebanu, we are able to further conclude that these three families constitute all the doubly primitive knots in $S^1 \times S^2$. Thus we bring the classification of lens space surgeries on knots in $S^1 \times S^2$ in line with the Berge Conjecture about lens space surgeries on knots in $S^3$.

Published 22 June 2016