Degree Date

2012

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Abstract

Legendrian knot theory is the study of topological knots and links that satisfy an additional, geometric condition. This condition, imposed by a contact structure, makes it possible for certain knots to be topologically equivalent without being Legendrian equivalent. In recent years, real-valued generating families and Morse theory have been used to define new invariants for Legendrian knots and links in jet spaces of the form J1(M, R), where M is a smooth, connected, closed manifold. This dissertation explores the extension of generating families to circle-valued functions, and their use in defining new invariants for Legendrian links in J1(M, S1). Two key results from the theory of real-valued generating families, the persistence and the uniqueness of certain generating families, are adapted to the case of circle-valued functions. These results are combined with ideas inspired by Morse-Novikov theory to establish a means of associating homology groups to a given three-component Legendrian link, from which we are able to define polynomial invariants. Finally, computations are performed to demonstrate how these invariants might eventually be applied; in particular, our computations suggest that the components of certain three-component Legendrian links cannot be permuted non-cyclically by Legendrian isotopy, even though such permutations are possible under topological isotopy.

Comments

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