#### Title

Virtual strings for closed curves with multiple components and filamentations for virtual links

#### Identifier

etd-07072005-121012

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### 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.

#### Date

2005

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Schellhorn, William, "Virtual strings for closed curves with multiple components and filamentations for virtual links" (2005). *LSU Doctoral Dissertations*. 709.

http://digitalcommons.lsu.edu/gradschool_dissertations/709

#### Committee Chair

Richard A. Litherland