Floer homology and knot complements

TL;DR

This paper introduces a new invariant of knot complements using Floer homology, providing tools to analyze surgeries on knots and calculating Floer homology for a class of 'perfect' knots, advancing knot theory and 3-manifold topology.

Contribution

It defines a filtered chain complex invariant of knot complements and develops methods to compute Floer homology for certain classes of knots, extending the Ozsvath-Szabo theory.

Findings

Defined the CF_r invariant for knot complements.

Derived information about surgeries using the exact triangle.

Calculated Floer homology for perfect knots.

Abstract

We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of \em{perfect} knots in S^3 for which CF_r has a particularly simple form. For these knots, formal properties of the Ozsvath-Szabo theory enable us to make a complete calculation of the Floer homology. This is the author's thesis; many of the results have been independently discovered by Ozsvath and Szabo in math.GT/0209056.