Doctor of Philosophy (PhD)
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.
Micklewright, Christopher. "Circle-Valued Generating Family Invariants." Ph.D. diss., Bryn Mawr College, 2012.