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Essak Seddiq

Knots and Links in Three-Manifolds

Teammate: Christopher Flippen

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: To prove that the quotient gordian and gordian-2 graphs are hyperbolic under different invariants, and possibly similar prove that one of those graphs takes the shape of a countably infinite complete graph.

A tip for others: Take it slow and steady. Be ready to learn completely new topics and ideas. Always think about the big picture when creating ideas and proofs, but be very strict and meticulous when writing them down so you can pinpoint any mistakes or fallacies.

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