#### Date of Award

5-2022

#### Document Type

Thesis

#### Degree Name

M.S.

#### Department

Mathematics

#### Committee Chair

Christine Lee, Ph.D.

#### Abstract

In this thesis we explore the relationship between quantum link invariants and the geometric and topological properties for a family of pretzel knots P(−2r, 2r + 1, 2r + s), r ≥ 1 and s ≥ 3, as predicted by the Slope conjecture. This conjecture asserts a connection between the degree of the colored Jones polynomial and boundary slopes of these knots. We introduce fundamentals of knot theory and prove that the minimum degree of the Jones polynomial for the family of pretzel knots P(−2r, 2r + 1, 2r + s), r ≥ 1 and s ≥ 3, is −6r − 3r + 3. In addition, we consider normal surface theory to find the boundary slopes of essential surfaces in the knot complement. As example, we consider the Lens Space L3,1 and compute the Q-matching equations associated with the triangulation. Since every essential surface is isotopic to a Haken sum of normal surfaces, we hope to one day combine the theorem proven in this thesis with normal surface theory in order to prove or disprove the Slope conjecture for the family of pretzel knots P(−2r, 2r + 1, 2r + s).

#### Recommended Citation

Swanepoel, Helene, "Slope Conjecture and Normal Surface Theory" (2022). *Theses and Dissertations*. 47.

https://jagworks.southalabama.edu/theses_diss/47