Longitude surgery on genus $1$ knots

Abstract:

Let $l(K)$ be the closed 3-manifold obtained by longitude surgery on the knot manifold K. Let C be the cube with holes obtained by removing an open regular neighborhood of a minimal spanning surface in K. The main result of this paper is that if K is of genus 1 and the longitude of K is in each term of the lower central series for ${\Pi _1}(C)$, then $l(K)$ is not homeomorphic to the connected sum of ${S^1} \times {S^2}$ and a homotopy 3-sphere. In particular, this implies we cannot obtain the connected sum of ${S^1} \times {S^2}$ and a homotopy 3-sphere by longitude surgery on any pretzel knot of genus 1.