## 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

#### Boston College, U.S.A.   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak6aecb2002c4b5c0c09ba6f2a5d6f1b4d').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy6aecb2002c4b5c0c09ba6f2a5d6f1b4d = 't&#97;&#111;l&#105;' + '&#64;'; addy6aecb2002c4b5c0c09ba6f2a5d6f1b4d = addy6aecb2002c4b5c0c09ba6f2a5d6f1b4d + 'bc' + '&#46;' + '&#101;d&#117;'; var addy_text6aecb2002c4b5c0c09ba6f2a5d6f1b4d = 't&#97;&#111;l&#105;' + '&#64;' + 'bc' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak6aecb2002c4b5c0c09ba6f2a5d6f1b4d').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy6aecb2002c4b5c0c09ba6f2a5d6f1b4d + '\'>'+addy_text6aecb2002c4b5c0c09ba6f2a5d6f1b4d+'<\/a>';

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.

Joint work with Yoav Moriah (Technion, Israel) and Tali Pinsky (Technion, Israel).