|Affiliation||University of Quebec at Montreal|
|Time||Jan 10 14:50~15:20|
|Title||The primitive/primitive and primitive/Seifert knot in $S^3$.|
Let $k$ be a simple closed curve in a genus two Heegaard surface $\Sigma$ of $S^3$ bounding handlebodies $H$ and $H'$. $k$ is called a primitive/primitive or double-primitive curve if adding a 2-handle to $H$ and $H'$ yields a solid torus. Similarly $k$ is called a primitive/Seifert curve if adding a 2-handle to ,say, $H$ and $H'$ yields a solid torus and a Seifert-fibered space respectively. Primitive/primitive and primitive/Seifert curves are of some interest because they have Dehn surgeries which yield lens spaces and Seifert-fibered spaces respectively. In this talk, I will explain how to find all primitive/primitive and primitive/Seifert knots in $S^3$ and how these have been grouped into the complete list of all such knots. The main tool for the classification uses R-R diagrams together with the fact that if adding a 2-handle to a genus two handlebody $H$ along a nonspearating curve $R$ on $\partial H$ embeds in $S^3$ as a knot exterior, then the meridian of the knot exterior can be obtained by surgery on $R$ along a wave. This is joint work with John Berge.