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

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.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Applied Mathematics

Grantor

Mathematics

Year dc:date.available

2005

Author dc:creator

• Schellhorn, William

• filamentations
• closed curves
• virtual strings
• virtual links

Statement dc:rights

• unrestricted
• Release the entire work immediately for access worldwide.

Identifier

etd-07072005-121012
https://repository.lsu.edu/gradschool_dissertations/709

OAI identifier oai:identifier

oai:repository.lsu.edu:gradschool_dissertations-1708