Donaldson's theorem, Heegaard Floer homology, and results on knots

Several interesting questions in knot theory reduce to the problem of determining whether a 3-manifold naturally associated to a knot can bound a specific type of 4-manifold. In this setting, two obstructions have come to bear quite strongly: one stems from a celebrated result of Simon Donaldson, and the other from a collection of numerical invariants defined by Peter Ozsvath and Zoltan Szabo in their Heegaard Floer homology theory. In this thesis, we describe a useful way to combine these two in order to obtain a finer obstruction of a lattice-theoretic nature, and apply it to several concrete problems. Specifically, we use it to give obstructions to a knot (1) possessing a lens space surgery; (2) being slice; (3) having unknotting number one; and (4) being quasi-alternating. In each case, we give specific applications: (1) we give an upper bound on the genus of a knot admitting a lens space surgery in terms of the surgery coefficient; (2) we determine the concordance orders of the odd 3-stranded pretzel knots; (3) we determine the alternating 3-braid knots with unknotting number one; and (4) we give the first examples of homologically thin links that are not quasi-alternating, and determine which pretzel links are quasi-alternating.