In this thesis we generalise three theorems from the literature on Heegaard Floer homology and Dehn surgery: one by Ozsv ́ath and Szab ́o on deficiency symmetries in half-integral L -space surgeries, and two by Greene which use Donaldson’s diagonali- sation theorem as an obstruction to integral and half-integral L -space surgeries. Our generalisation is two-fold: first, we eliminate the L -space conditions, opening these techniques up for use with much more general 3-manifolds, and second, we unify the integral and half-integral surgery results into a broader theorem applicable to non- zero rational surgeries in S 3 which bound sharp, simply connected, negative-definite smooth 4-manifolds. Such 3-manifolds are quite common and include, for example, a huge number of Seifert fibred spaces. Over the course of the first three chapters, we begin by introducing background material on knots in 3-manifolds, the intersection form of a simply connected 4- manifold, Spin- and Spin c -structures on 3- and 4-manifolds, and Heegaard Floer ho- mology (including knot Floer homology). While none of the results in these chapters are original, all of them are necessary to make sense of what follows. In Chapter 4, we introduce and prove our main theorems, using arguments that are predominantly algebraic or combinatorial in nature. We then apply these new theorems to the study of unknotting number in Chapter 5, making considerable headway into the extremely difficult problem of classifying the 3-strand pretzel knots with unknotting number one. Finally, in Chapter 6, we present further applications of the main theorems, ranging from a plan of attack on the famous Seifert fibred space realisation problem to more biologically motivated problems concerning rational tangle replacement. An appendix on the implications of our theorems for DNA topology is provided at the end.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:693933 |
| Date | January 2013 |
| Creators | Gibbons, Julien Charles |
| Contributors | Buck, Dorothy |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/39364 |
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