Longitude surgery on genus 1 knots

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Abstract
Let l ( K ) 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 Π 1 ( C ) {\Pi _1}(C) , then l ( K ) l(K) is not homeomorphic to the connected sum of S 1 × S 2 {S^1} \times {S^2} and a homotopy 3-sphere. In particular, this implies we cannot obtain the connected sum of S 1 × S 2 {S^1} \times {S^2} and a homotopy 3-sphere by longitude surgery on any pretzel knot of genus 1.

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