Tunnel number one knots satisfy the Berge Conjecture

TL;DR

This paper proves that tunnel number one knots in certain 3-manifolds that admit lens space surgeries are doubly primitive, confirming the Berge Conjecture for these cases and resolving related conjectures in specific manifolds.

Contribution

It establishes the Berge Conjecture for tunnel number one knots in $S^3$ and extends results to knots in $S^2 imes S^1$, confirming conjectures by Greene and Baker-Buck-Lecuona.

Findings

Tunnel number one knots with lens space surgeries are doubly primitive.

The Berge Conjecture is proved for knots in $S^3$.

Conjectures for knots in $S^2 imes S^1$ are confirmed.

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.