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      research:

On Fox's Conjecture of Quasi-Alternating Knots, We verify Fox's trapezoidal conjecture for a family of alternating pretzel knots. We also extend the conjecture to quasi-alternating slice knots and verify the conjecture for a class of these knots. Summer 2022

An Exploration of the Geometry of Hyperbolic Maps, We define an analog of polar coordinates for the hyperbolic plane and we propose a definition for the power function on the hyperbolic numbers. Calculus on the hyperbolic numbers allows the solution of the wave equation in much the same fashion as calculus on the complex numbers allows the solution to Laplace's Equation. In complex analysis, the technique of conformal mapping is used to create solutions of Laplace's equation with suitable boundary conditions. We study mappings of the hyperbolic plane with the hope of developing a similar technique to create wave equation solutions. Summer 2021