Type of Document	Dissertation
Author	Schellhorn, William
URN	etd-07072005-121012
Title	Virtual Strings for Closed Curves with Multiple Components and Filamentations for Virtual Links
Degree	Doctor of Philosophy (Ph.D.)
Department	Mathematics
Advisory Committee	Advisor Name Title Richard A. Litherland Committee Chair Frank Neubrander Committee Member Lawrence Smolinsky Committee Member Oliver Dasbach Committee Member Robert Perlis Committee Member Kemin Zhou Dean's Representative
Keywords	filamentations closed curves virtual strings virtual links
Date of Defense	2005-04-22
Availability	unrestricted
Abstract The theory of filaments on oriented chord diagrams can be used to detect some non-classical virtual knots. We extend existing filament techniques to virtual links with more than one component and give examples of virtual links that these techniques can detect as non-classical. Given a signed Gauss word underlying an oriented chord diagram, we describe how to construct a finite sequence of integers that encodes all of the filament information for the diagram. We also introduce a square array of integers called a MIN-square that summarizes the filament information about all of the signed Gauss words having a given Gauss word shape. 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. 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 of their alphabet sets. In addition, we prove that the unordered triple consisting of the Gauss paragraph, the word-wise partition, and a related word-wise partition associated to a closed curve on the 2-sphere is a full homeomorphism invariant of the closed curve. We conclude by introducing a multi-variable polynomial that is a homotopy invariant of virtual strings with multiple circles.
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