{"id":{"repo_id":"lsu-thes","oai_identifier":"oai:repository.lsu.edu:gradschool_dissertations-1551"},"canonical_url":"https://search.dev.ndltd.org/etd/lsu-thes/oai:repository.lsu.edu:gradschool_dissertations-1551","repository":{"repo_id":"lsu-thes","name":"Lousiana State University","base_url":"https://repository.lsu.edu/do/oai/"},"display":{"title":"Hypercube diagrams for knots, links, and knotted tori","abstract":"For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. There is also a Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number. Finally, there is a generalization of cube diagrams, called hypercube diagrams. We use such diagrams, which represent immersed Lagrangian tori in R^4 to study embedded Legendrian tori in the standard contact space. We then show how to compute one of the classical invariants, the rotation class, and discuss applications to contact homology.","abstract_html":"For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. There is also a Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number. Finally, there is a generalization of cube diagrams, called hypercube diagrams. We use such diagrams, which represent immersed Lagrangian tori in R^4 to study embedded Legendrian tori in the standard contact space. We then show how to compute one of the classical invariants, the rotation class, and discuss applications to contact homology.","abstract_has_math":false,"creators":["McCarty, Ben"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied 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":"2026-07-24T02:57:49Z","subjects":["knots","links","contact","Legendrian"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-05252012-130050","https://repository.lsu.edu/gradschool_dissertations/552"],"render_values":[{"text":"etd-05252012-130050","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/552","href":"https://repository.lsu.edu/gradschool_dissertations/552","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.552","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["McCarty, Ben"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-05-01"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:09:12Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["knots","links","contact","Legendrian"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-05252012-130050","10.31390/gradschool_dissertations.552","https://repository.lsu.edu/gradschool_dissertations/552"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. There is also a Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number. Finally, there is a generalization of cube diagrams, called hypercube diagrams. We use such diagrams, which represent immersed Lagrangian tori in R^4 to study embedded Legendrian tori in the standard contact space. We then show how to compute one of the classical invariants, the rotation class, and discuss applications to contact homology."]},{"key":"dc:title","label":"Title","values":["Hypercube diagrams for knots, links, and knotted tori"]}]}],"canonical_facts":{"dc:creator":["McCarty, Ben"],"dc:date":["2012-05-01"],"dc:date.available":["2022-05-12T23:09:12Z"],"dc:description.abstract":["For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. There is also a Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number. Finally, there is a generalization of cube diagrams, called hypercube diagrams. We use such diagrams, which represent immersed Lagrangian tori in R^4 to study embedded Legendrian tori in the standard contact space. We then show how to compute one of the classical invariants, the rotation class, and discuss applications to contact homology."],"dc:identifier":["etd-05252012-130050","10.31390/gradschool_dissertations.552","https://repository.lsu.edu/gradschool_dissertations/552"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["knots","links","contact","Legendrian"],"dc:title":["Hypercube diagrams for knots, links, and knotted tori"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:57:49Z"}