Department of Mathematics
Studying 3-manifolds using knots
Michael McLendon
Washington College, Chestertown, Maryland
The skein module of a 3-manifold is an algebraic object formed by the types of knots and links that the manifold can contain. In the words of Jozef Przytycki, skein theory is ``algebraic topology based on knots.'' We will look at the skein module of a 3-manifold when the manifold is defined via a Heegaard splitting, M = H0 UF H1. Here H0 and H1 are two solid handlebodies glued together to form M along their common boundary surface F. Using this Heegaard splitting of M in conjunction with the skein modules of H0, H1, and F, we can define an indexed list of modules called the Hochschild homology of the Heegaard splitting. The zeroth Hochschild homology recovers the information in the skein module of the manifold and the higher Hochschild homology modules may provide additional information about the manifold.
All interested persons are welcome.