**Faculty mentor:** Dr. Allison Moore

**Department:** Mathematics

**In a nutshell:** Our project involves the study of knots and how we can describe and distinguish different kinds of knots by attributing mathematical quantities to identifiable features of the knots, such as the number of twists a given knot has. We put knots into different groups according to the value they hold for the given mathematical quantity we are measuring. We determine which groups are related to each other by seeing if there is a knot in one group that can be physically transformed into a knot in another group, with specific restrictions on what transformation we can make in the knot, such as undoing only one twist.

**In a bigger shell:** The goal of this paper is to study quotients of the regular and H2-Gordian graph under integer-valued invariants in order to determine which invariants result in hyperbolic quotient graphs. These invariants stem from the Jones polynomial including the determinant, tricolorability number, and span. We use properties of the connected sum and of special families of knots including generalized twist knots and T(2,n) torus knots to prove the hyperbolicity of these quotient graphs. We also describe the structure of these quotient graphs and relate them to well-known families of countably infinite graphs such as the complete graph and the path graph.

**End of year goal:** Our goal is to prove the hyperbolicity of several quotient graphs of the Gordian graph and the H2-Gordian graph.

**A tip for others:** Be ready to learn new ways of thinking about mathematical concepts. It is important to be willing to think about the same problem multiple ways, especially if one method of solving it ends up not working. If a problem seems impossible to solve, sometimes asking a different question is important, or rephrasing the question.