Floer homology and knot complements

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.

83 pages; Harvard thesis

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• Heegaard Floer homology and alternating knots
• The support of the Khovanov's invariants for alternating knots
• Holomorphic triangles and invariants for smooth four-manifolds
• Holomorphic disks and knot invariants
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