Publications and Preprints
Undergraduate
A Theoretical Schema for Building Weavings of Nets via Colored Tilings of TwoDimensional Spaces and Some Simple Polyhedral, Planar and ThreePeriodic Examples (with Stephen Hyde, 13 pages, submitted Jan 2018, IJC 589 (2018) 11441156) We analyse twocomponent “weavings” made of a pair of dual (p,q) and (q,p) nets that undulate on both sides of the sphere, the plane and the hyperbolic plane. Families of weavings are described, sharing a common parent net. The examples describe zero, two and threeperiodic weavings in threespace. We derive all edge2transitive weavings with (p,q) = (3,3), (4,3), (4,4), (4,6) using DelaneyDress tiling theory, described in detail. The twodimensional hyperbolic weaving are mapped into (euclidean) threespace to form a pair of catenated crystalline nets. The examples suggest generalisations to other weavings on surfaces, including weavings of filaments. A simple hyperbolic weaving of filaments is derived, analogous to the common warpandweft filament weaving in the plane. The resulting threeperiodic pattern is related to the molecularscale weaving in the synthetic COF505 material synthesized by Liu et al (2016). 

Khovanov complexes of rational tangles (20 pages, posted Jan 2017, updated Dec 2018, arXiv:1701.07525) We show that the Khovanov complex of a rational tangle has a very simple representative whose backbone of nonzero morphisms forms a zigzag. Furthermore, this minimal complex can be computed quickly by an inductive algorithm. (For example, we calculate Kh(8_{2}) by hand.) We find that the bigradings of the subobjects in these minimal complexes can be described by matrix actions, which after a change of basis is the reduced Burau representation of B_{3}. 