Research

My research is in knot theory and low-dimensional topology, with a particular focus on knot mosaics, which provide combinatorial models of knots using prescribed tiles arranged in an array. My current work studies spherical knot mosaics, which adapt these constructions to a spherical setting and investigate how classical knot invariants interact with invariants arising from mosaic representations.

• "Face Efficient Spherical Knot Mosaics," joint work with Samantha Pezzimenti   and Peyton Wood. In preparation

TLDR: We introduce face-efficient spherical knot n-mosaics, which minimize the number of faces in a spherical mosaic representation of a knot, construct such mosaics for all knots with crossing number ≤ 8 when sm(K)≤n<m(K) and establish bounds for knots with higher crossing numbers.

• "Spherical Knot Mosaics," joint work with Peyton Wood. Submitted. You can view the preprint here 

TLDR: We introduce spherical knot mosaics, which represent knots by tiling the surface of a sphere, define several new knot and link invariants from this framework, and show how they both improve upon classical knot mosaics and yield bounds relating to classical knot invariants.