Elliptic Involution in Bordered Heegaard Floer Homology

Abstract

In this thesis, we study the elliptic involution from the point of view of the bordered Heegaard Floer homology. We show that the bordered Heegaard Floer invariant CFD^hat of a knot complement in S^3 is invariant under the elliptic involution on its boundary. As a computational result, we demonstrate how the bordered Heegaard Floer invariant CFDA^hat associated to the elliptic involution affects such CFD^hat complexes via A infinity tensor product. Globally, it "flips" the entire complex, while locally, it modifies the complex nicely in an arrow-wise fashion. The latter local description can be extended to quarter boundary Dehn twists as well. Additionally, the Heegaard Floer homology of the associated open book decomposition to the elliptic involution is computed.