Tunnel number one knots satisfy the Berge Conjecture

Abstract

Let $K$ be a tunnel number one knot in $M$ with irreducible knot exterior, where $M$ is either $S^3$, or a connected sum of $S^2\times S^1$ with any lens space. (In particular, this includes $M=S^2\times S^1$.) We prove that if a non-trivial Dehn surgery on $K$ yields a lens space, then $K$ is a doubly primitive knot in $M$. For $M=S^3$ this resolves the tunnel number one Berge Conjecture. For $M=S^2\times S^1$ this resolves a conjecture of Greene and Baker-Buck-Lecuona for tunnel number one knots.

72 pages, 38 figures. Geometry & Topology, to appear

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