# Combinatorial Legendrian knot invariants : representation numbers

Cardinal Scholar
 dc.contributor.advisor Rutherford, Dan (Daniel Robert) dc.contributor.author Murray, Justin A. dc.date.accessioned 2019-05-13T17:35:18Z dc.date.available 2019-05-13T17:35:18Z dc.date.issued 2019-05-04 dc.identifier.uri http://cardinalscholar.bsu.edu/handle/123456789/201703 dc.description.abstract The study of Legendrian knots lies within the larger elds of contact geometry and knot en_US theory. The requirements for Legendrian invariance is strictly stronger than its topological analog, as there are Legendrian knots that are not Legendrian isotopic, but are isotopic as topological knots. As with topological knot theory, the classi cation problem, i.e. classify all knots up to Legendrian isotopy, is still a main problem in Legendrian knot theory. We consider Legendrians lying within the standard contact structure (R3; std). One of the most powerful Legendrian knot invariants is a di erential graded algebra, (A; @), introduced by Chekanov and Eliashberg. It has been shown that representation numbers, a normal- ized count of representations from (A; @), are a Legendrian knot invariant. This project addresses the Chekanov-Eliashberg di erential graded algebra and representation numbers, and provides a de nition for the 1-graded 2-colored ruling polynomials R1 2;K(q). We then show that R1 2;K(q) recovers the 1-graded total 2-dimensional representation number. dc.description.sponsorship Department of Mathematical Sciences dc.description.tableofcontents Legendrian knots and classical invariants -- The Chekanov-Eliashberg DGA -- Representation numbers -- Representations of a satellite dc.subject.lcsh Knot theory. dc.subject.lcsh Differential algebra. dc.subject.lcsh Representations of algebras. dc.title Combinatorial Legendrian knot invariants : representation numbers en_US dc.description.degree Thesis (M.S.) en_US
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• Master's Theses [5454]
Master's theses submitted to the Graduate School by Ball State University master's degree candidates in partial fulfillment of degree requirements.