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.
","abstract_html":"<p>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 J<sup>1</sup>(<em>M</em>, R), where<em> M</em> 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 J<sup>1</sup>(<em>M, S</em><sup>1</sup>). 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.</p>","abstract_has_math":false,"creators":["Micklewright, Christopher"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Bryn Mawr only","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T01:23:36Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://repository.brynmawr.edu/dissertations/76","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Micklewright, Christopher"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2013-07-24T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Bryn Mawr only"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://repository.brynmawr.edu/dissertations/76"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Circle-Valued Generating Family Invariants"]}]}],"canonical_facts":{"dc:creator":["Micklewright, Christopher"],"dc:date.available":["2013-07-24T07:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://repository.brynmawr.edu/dissertations/76"],"dc:subject":["Mathematics"],"dc:title":["Circle-Valued Generating Family Invariants"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Bryn Mawr only"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:23:36Z"}