William J. Schellhorn

arXiv2006

Filamentations for Virtual Links

In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.

math.GT↗

arXiv2005

Virtual Strings for Closed Curves with Multiple Components

A Gauss paragraph is a combinatorial formulation of a generic closed curve with multiple components on some surface. A virtual string is a collection of circles with arrows that represent the crossings of such a curve. Every closed curve has an underlying virtual string and every virtual string has an underlying Gauss paragraph. A word-wise partition is a partition of the alphabet set of a Gauss paragraph that satisfies certain conditions with respect to the Gauss paragraph. In this paper we use the theory of virtual strings to obtain a combinatorial description of closed curves in the 2-sphere (and therefore 2-dimensional Euclidean space) in terms of Gauss paragraphs and word-wise partitions.

math.GT↗