#### Date of Graduation

5-2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Degree Level

Graduate

#### Department

Mathematical Sciences

#### Advisor

Jeremy Van Horn-Morris

#### Committee Member

Allan Cochran

#### Second Committee Member

Chaim Goodman-Strauss

#### Third Committee Member

Yo'av Rieck

#### Keywords

Pure sciences, Arc index, Grid diagrams, Rectangular diagrams, Thurston-bennequin number

#### 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.

#### Citation

Thomas, E. G.
(2016). The Maximal Thurston-Bennequin Number on Grid Number n Diagrams. * Graduate Theses and Dissertations*
Retrieved from https://scholarworks.uark.edu/etd/1540