We will prove an upper bound for the Thurston-Bennequin number of Legendrian knots and links on a rectangular grid with arc index n.
TB(n)=CR(n)-[n/2]
In order to prove the bound, we will separate our work for when n is even and when n is odd. After we prove the upper bound, we will show that there are unique knots and links on each grid which achieve the upper bound. When n is even, torus links achieve the maximum, and when n is odd, torus knots achieve the maximum.
","abstract_html":"<p>We will prove an upper bound for the Thurston-Bennequin number of Legendrian knots and links on a rectangular grid with arc index n.</p> <p>TB(n)=CR(n)-[n/2]</p> <p>In order to prove the bound, we will separate our work for when n is even and when n is odd. After we prove the upper bound, we will show that there are unique knots and links on each grid which achieve the upper bound. When n is even, torus links achieve the maximum, and when n is odd, torus knots achieve the maximum.</p>","abstract_has_math":false,"creators":["Thomas, Emily Goins"],"institution":null,"degree_name":"Doctor of Philosophy in Mathematics (PhD)","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Cochran, Allan","Goodman-Strauss, Chaim"],"advisors":["Van Horn-Morris, Jeremy"],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-05-01T07:00:00Z","date_published":"2016-05-01T07:00:00Z","updated_at":"2026-07-24T00:59:01Z","subjects":["Pure sciences","Arc index","Grid diagrams","Rectangular diagrams","Thurston-bennequin number","Other Applied Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.uark.edu/etd/1540","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Cochran, Allan","Goodman-Strauss, Chaim"]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Van Horn-Morris, Jeremy"]},{"key":"dc:creator","label":"Author","values":["Thomas, Emily Goins"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2016"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-09-29T07:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy in Mathematics (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Arc index","Grid diagrams","Rectangular diagrams","Thurston-bennequin number","Other Applied Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.uark.edu/etd/1540"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We will prove an upper bound for the Thurston-Bennequin number of Legendrian knots and links on a rectangular grid with arc index n.
TB(n)=CR(n)-[n/2]
In order to prove the bound, we will separate our work for when n is even and when n is odd. After we prove the upper bound, we will show that there are unique knots and links on each grid which achieve the upper bound. When n is even, torus links achieve the maximum, and when n is odd, torus knots achieve the maximum.
"]},{"key":"dc:title","label":"Title","values":["The Maximal Thurston-Bennequin Number on Grid Number n Diagrams"]}]}],"canonical_facts":{"dc:contributor":["Cochran, Allan","Goodman-Strauss, Chaim"],"dc:contributor.advisor":["Van Horn-Morris, Jeremy"],"dc:creator":["Thomas, Emily Goins"],"dc:date":["2016"],"dc:date.available":["2017-09-29T07:00:00Z"],"dc:description.abstract":["We will prove an upper bound for the Thurston-Bennequin number of Legendrian knots and links on a rectangular grid with arc index n.
TB(n)=CR(n)-[n/2]
In order to prove the bound, we will separate our work for when n is even and when n is odd. After we prove the upper bound, we will show that there are unique knots and links on each grid which achieve the upper bound. When n is even, torus links achieve the maximum, and when n is odd, torus knots achieve the maximum.
"],"dc:identifier":["https://scholarworks.uark.edu/etd/1540"],"dc:subject":["Pure sciences","Arc index","Grid diagrams","Rectangular diagrams","Thurston-bennequin number","Other Applied Mathematics"],"dc:title":["The Maximal Thurston-Bennequin Number on Grid Number n Diagrams"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy in Mathematics (PhD)"]},"updated_at":"2026-07-24T00:59:01Z"}