Date of Graduation

12-2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

Jeremy Van Horn-Morris

Committee Member

Yo'av Rieck

Second Committee Member

Matthew Clay

Third Committee Member

Matthew Day

Keywords

Grid Diagrams, Knots, Legendrian, Singular Links

Abstract

If $\Lambda_{1}^{\ast}$ and $\Lambda_{2}^{\ast}$ are two oriented singular Legendrian links that are Legendrian isotopic, we first construct front diagram representations of $\Lambda_{1}^{\ast}$ and $\Lambda_{2}^{\ast}$ that have a natural allowable singular gird diagram associated to them. These allowable singular grid diagrams will always correspond to singular Legendrian links. The grid Legendrian invariants, $\lambda^{\pm}$, in the nonsingular grid homology theory have a natural extension to the singular grid theory, and are natural under the newly defined singular grid moves. This gives an invariant of singular Legendrian links, and in fact, a broader class of singular links.

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