Sahamie, Bijan
(2009).
Dehn Twists and Heegaard Floer Homology.
PhD thesis, Universität zu Köln.
Abstract
We derive a representation of the hatversion of Heegaard Floer homology in case we change the associated Heegaard diagram with a Dehn twist. Result of this description is a new exact sequence in the hat-version of Heegaard Floer homology. To give the involved modules a suitable geometric interpretation, we generalize the knot Floer homology HFK to homologically non-trivial knots and relax the admissibility conditions used in their definition. As part of the exact sequence we obtain a map, which we show not to depend on the choices made in its definition, but on the cobordism induced by the Dehn twist. With this map we derive a naturality property between the invariant of Legendrian knots \loss and the contact element and give three applications. Finally, we investigate the relationship between the newly defined exact sequences and the well-known surgery exact triangle in knot Floer homology. With a suitable modification of the construction process of the surgery exact triangle we derive a strong relationship to the newly defined exact sequences. This, finally, results in a relationship between counting holomorphic triangles in doubly-pointed Heegaard triple diagrams and counting holomorphic discs in pointed Heegaard diagrams.
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