Publications and Preprints

A Theoretical Schema for Building Weavings of Nets via Colored Tilings of Two-Dimensional Spaces and Some Simple Polyhedral, Planar and Three-Periodic Examples

(with Stephen Hyde, 13 pages, submitted Jan 2018, IJC 58-9 (2018) 1144-1156)

We analyse two-component “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 three-periodic weavings in three-space. We derive all edge-2-transitive weavings with (p,q) = (3,3), (4,3), (4,4), (4,6) using Delaney-Dress tiling theory, described in detail. The two-dimensional hyperbolic weaving are mapped into (euclidean) three-space 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 warp-and-weft filament weaving in the plane. The resulting three-periodic pattern is related to the molecular-scale weaving in the synthetic COF-505 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 non-zero morphisms forms a zig-zag. Furthermore, this minimal complex can be computed quickly by an inductive algorithm. (For example, we calculate Kh(8₂) 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 B₃.