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Full-Text Articles in Physical Sciences and Mathematics
Thickened Surfaces, Checkerboard Surfaces, And Quantum Link Invariants, Joseph W. Boninger
Thickened Surfaces, Checkerboard Surfaces, And Quantum Link Invariants, Joseph W. Boninger
Dissertations, Theses, and Capstone Projects
This dissertation has two parts, each motivated by an open problem related to the Jones polynomial. The first part addresses the Volume Conjecture of Kashaev, Murakami, and Murakami. We define a polynomial invariant, JTn, of links in the thickened torus, which we call the nth toroidal colored Jones polynomial, and we show JTn satisfies many properties of the original colored Jones polynomial. Most significantly, JTn exhibits volume conjecture behavior. We prove a volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions …
Intercusp Geodesics And Cusp Shapes Of Fully Augmented Links, Rochy Flint
Intercusp Geodesics And Cusp Shapes Of Fully Augmented Links, Rochy Flint
Dissertations, Theses, and Capstone Projects
We study the geometry of fully augmented link complements in the 3-sphere by looking at their link diagrams. We extend the method introduced by Thistlethwaite and Tsvietkova to fully augmented links and define a system of algebraic equations in terms of parameters coming from edges and crossings of the link diagrams. Combining it with the work of Purcell, we show that the solutions to these algebraic equations are related to the cusp shapes of fully augmented link complements. As an application we use the cusp shapes to study the commensurability classes of fully augmented links.