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`http://arks.princeton.edu/ark:/88435/dsp01dj52w6911`

Title: | Floergåsbord |

Authors: | Lewallen, Sam Jay |

Advisors: | Szabo, Zoltan |

Contributors: | Mathematics Department |

Keywords: | Floer homology Geometry Knot theory Topological quantum field theory Topology |

Subjects: | Mathematics |

Issue Date: | 2014 |

Publisher: | Princeton, NJ : Princeton University |

Abstract: | In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot K in a closed, oriented 3-manifold M, we use SU(2) representation spaces and the Lagrangian field theory framework of Wehrheim and Woodward to define a new homological knot invariant S(K). We then use a result of Ivan Smith to show that when K is a (1,1) knot in S^3 (a set of knots which includes torus knots, for example) the rank of S(K) otimes C agrees with the rank of knot Floer homology, hat{HFK}(K) otimes C, and we conjecture that this holds in general for any knot K. In Chapter 3, we prove a somewhat strange result, giving a purely topological formula for the Jones polynomial of a 2-bridge knot K susbet S^3. First, for any lens space L(p,q), we combine the d-invariants from Heegaard Floer homology with certain Atiyah-Patodi-Singer/Casson-Gordon &rho-invariants to define a function Ip,q : Z/pZ → Z Let K = K(p,q) denote the 2-bridge knot in S^3 whose double-branched cover is L(p,q), let sigma(K) denote the knot signature, and let O denote the set of relative orientations of K, which has cardinality 2^{(# of components of K)-1}. Then we prove the following formula for the Jones polynomial J(K): i^-sigma(K) q^(3sigma(K)) J(K) = \sum_{o \in O} (iq)^{2sigma(K^o)} + q^(-1) - q^1 \sum_{s \in Z/pZ} (iq)^(I_{p,q}(s)) (here, i = sqrt{-1}). In Chapter 4, we present joint work with Adam Levine, concerning Heegaard Floer homology and the orderability of fundamental groups. Namely, we prove that if hat{CF}(M) is particularly simple, i.e., M is what we call a "strong L-space," then \pi_1(M) is not left-orderable. |

URI: | http://arks.princeton.edu/ark:/88435/dsp01dj52w6911 |

Alternate format: | The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog |

Type of Material: | Academic dissertations (Ph.D.) |

Language: | en |

Appears in Collections: | Mathematics |

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Lewallen_princeton_0181D_11030.pdf | 441.83 kB | Adobe PDF | View/Download |

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