## View abstract

### Session S28 - Knots, Surfaces, 3-manifolds

Thursday, July 15, 18:40 ~ 19:10 UTC-3

## Berge Conjecture for tunnel number one knots

### Tao Li

Let $K$ be a tunnel number one knot in $M$, where $M$ is either $S^3$, $S^2\times S^1$, or a connected sum of $S^2\times S^1$ with a lens space. We prove that if a 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.