{"id":{"repo_id":"wustl","oai_identifier":"oai:openscholarship.wustl.edu:etd-1146"},"canonical_url":"https://search.dev.ndltd.org/etd/wustl/oai:openscholarship.wustl.edu:etd-1146","repository":{"repo_id":"wustl","name":"Washington University in St. Louis","base_url":"https://openscholarship.wustl.edu/do/oai/"},"display":{"title":"Connections between Floer-type invariants and Morse-type invariants of Legendrian knots.","abstract":"We investigate existing Legendrian knot invariants and discover new connections between the theory of generating families, normal rulings and the Chekanov-Eliashberg differential graded algebra: CE-DGA). Given a Legendrian knot $\\sK$ with generic front projection $\\sfront$, we define a combinatorial/algebraic object on $\\sfront$ called a \\emph{Morse complex sequence}, abbreviated MCS. An MCS encodes a finite sequence of Morse homology complexes. Every suitably generic generating family for $\\sfront$ admits an MCS and every MCS has a naturally associated graded normal ruling. In addition, every MCS has a naturally associated augmentation of the CE-DGA of the Ng resolution $\\sNgres$ of the front $\\sfront$. In this manner, an MCS connects generating families, normal rulings and augmentations. We place an equivalence relation on the set $\\sDMCS$ of MCSs on $\\sfront$ and prove that there exists a natural surjection from the equivalence classes of $\\sDMCS$, denoted $\\sDMCSeq$, to the set of chain homotopy classes of augmentations of $\\sNgres$, denoted $\\sAugNgresch$. In the case of Legendrian isotopy classes admitting representatives with two-bridge front projections, $\\sDMCSeq$ and $\\sAugNgresch$ are in bijection.","abstract_html":"We investigate existing Legendrian knot invariants and discover new connections between the theory of generating families, normal rulings and the Chekanov-Eliashberg differential graded algebra: CE-DGA). Given a Legendrian knot $\\sK$ with generic front projection $\\sfront$, we define a combinatorial/algebraic object on $\\sfront$ called a \\emph{Morse complex sequence}, abbreviated MCS. An MCS encodes a finite sequence of Morse homology complexes. Every suitably generic generating family for $\\sfront$ admits an MCS and every MCS has a naturally associated graded normal ruling. In addition, every MCS has a naturally associated augmentation of the CE-DGA of the Ng resolution $\\sNgres$ of the front $\\sfront$. In this manner, an MCS connects generating families, normal rulings and augmentations. We place an equivalence relation on the set $\\sDMCS$ of MCSs on $\\sfront$ and prove that there exists a natural surjection from the equivalence classes of $\\sDMCS$, denoted $\\sDMCSeq$, to the set of chain homotopy classes of augmentations of $\\sNgres$, denoted $\\sAugNgresch$. In the case of Legendrian isotopy classes admitting representatives with two-bridge front projections, $\\sDMCSeq$ and $\\sAugNgresch$ are in bijection.","abstract_has_math":true,"creators":["Henry, Michael"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Rachel Roberts"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-01-01T08:00:00Z","date_published":"2009-01-01T08:00:00Z","updated_at":"2026-07-24T06:12:48Z","subjects":["Mathematics","contact topology","knot theory","Legendrian knot theory","Low-dimensional topology","Topology"],"languages":["English (en)"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.7936/K71G0JBG"],"render_values":[{"text":"https://doi.org/10.7936/K71G0JBG","href":"https://doi.org/10.7936/K71G0JBG","code":true}]}]},"links":{"outbound_url":"https://openscholarship.wustl.edu/etd/147","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Rachel Roberts"]},{"key":"dc:creator","label":"Author","values":["Henry, Michael"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2010-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"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","contact topology","knot theory","Legendrian knot theory","Low-dimensional topology","Topology"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English (en)"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://openscholarship.wustl.edu/etd/147"]},{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.7936/K71G0JBG"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We investigate existing Legendrian knot invariants and discover new connections between the theory of generating families, normal rulings and the Chekanov-Eliashberg differential graded algebra: CE-DGA). Given a Legendrian knot $\\sK$ with generic front projection $\\sfront$, we define a combinatorial/algebraic object on $\\sfront$ called a \\emph{Morse complex sequence}, abbreviated MCS. An MCS encodes a finite sequence of Morse homology complexes. Every suitably generic generating family for $\\sfront$ admits an MCS and every MCS has a naturally associated graded normal ruling. In addition, every MCS has a naturally associated augmentation of the CE-DGA of the Ng resolution $\\sNgres$ of the front $\\sfront$. In this manner, an MCS connects generating families, normal rulings and augmentations. We place an equivalence relation on the set $\\sDMCS$ of MCSs on $\\sfront$ and prove that there exists a natural surjection from the equivalence classes of $\\sDMCS$, denoted $\\sDMCSeq$, to the set of chain homotopy classes of augmentations of $\\sNgres$, denoted $\\sAugNgresch$. In the case of Legendrian isotopy classes admitting representatives with two-bridge front projections, $\\sDMCSeq$ and $\\sAugNgresch$ are in bijection."]},{"key":"dc:title","label":"Title","values":["Connections between Floer-type invariants and Morse-type invariants of Legendrian knots."]}]}],"canonical_facts":{"dc:contributor":["Rachel Roberts"],"dc:creator":["Henry, Michael"],"dc:date.available":["2010-01-01T08:00:00Z"],"dc:description.abstract":["We investigate existing Legendrian knot invariants and discover new connections between the theory of generating families, normal rulings and the Chekanov-Eliashberg differential graded algebra: CE-DGA). Given a Legendrian knot $\\sK$ with generic front projection $\\sfront$, we define a combinatorial/algebraic object on $\\sfront$ called a \\emph{Morse complex sequence}, abbreviated MCS. An MCS encodes a finite sequence of Morse homology complexes. Every suitably generic generating family for $\\sfront$ admits an MCS and every MCS has a naturally associated graded normal ruling. In addition, every MCS has a naturally associated augmentation of the CE-DGA of the Ng resolution $\\sNgres$ of the front $\\sfront$. In this manner, an MCS connects generating families, normal rulings and augmentations. We place an equivalence relation on the set $\\sDMCS$ of MCSs on $\\sfront$ and prove that there exists a natural surjection from the equivalence classes of $\\sDMCS$, denoted $\\sDMCSeq$, to the set of chain homotopy classes of augmentations of $\\sNgres$, denoted $\\sAugNgresch$. In the case of Legendrian isotopy classes admitting representatives with two-bridge front projections, $\\sDMCSeq$ and $\\sAugNgresch$ are in bijection."],"dc:identifier":["https://openscholarship.wustl.edu/etd/147"],"dc:identifier.doi":["https://doi.org/10.7936/K71G0JBG"],"dc:language":["English (en)"],"dc:subject":["Mathematics","contact topology","knot theory","Legendrian knot theory","Low-dimensional topology","Topology"],"dc:title":["Connections between Floer-type invariants and Morse-type invariants of Legendrian knots."],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T06:12:48Z"}