Topology Seminar for 11/7/2002

Peter Ozsvath
Columbia University

Holomorphic disks and knot invariants

Abstract

I will describe some recent work with Zoltan Szabo in which we define Floer homology groups for knots. For knots in the three-sphere, the generators of the chain complex have an explicit combinatorial description in terms of the planar projection of the knot, although the differentials in general are quite elusive (counting holomorphic disks in symmetric powers of a Heegaard surface). However, this information already suffices to determine the knot homology of many knots, including all alternating knots, and several additional families. Applications of these results include: restrictions on the Alexander polynomials of alternating knots, obstructions for knots admitting Seifert fibered surgeries, and restrictions on the algebraic topology of smooth four-manifolds which bound $+1$-surgery along the knot.