Document Type

Dissertation

Date of Award

1975

Keywords

Knot theory, Algorithms, Computer algorithms

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Prabir Roy

Second Advisor

Louis F. McAuley

Third Advisor

Patricia McAuley

Abstract

The concern of this paper is with making accessible many different piecewise linear knot spaces which contain incompressible surfaces of arbitrarily high genus.

W. Haken in [3]* and F. Waldhausen in [7] discuss the use of incompressible surfaces in 3-manifolds to in some way characterize topological properties of the manifold. H. Schubert in [6] uses Haken’s theory of normal surfaces [2] to determine specific properties of certain 3-manifolds and the determinations are provided by the constructibility of what he calls characteristic surfaces, surfaces defined to be characteristic to the property under investigation. Which surfaces are characteristic depends upon the topological property of interest.

Some of the algorithms of Haken [3] and Waldhausen [7] require only that at each step an incompressible surface be available. Some of Schubert's algorithms [6] require that a surface of minimum genus be constructible at each step.

In this paper is developed a general sequence of specifications and theorems which prescribe necessary and sufficient conditions for certain surfaces in a type of knot space to be incompressible, and prescribe a structure on the knot space which is proved to assure these conditions. The general structure is then utilized to give several specific and different algorithms which prescribe many immediate constructions of knot spaces which contain incompressible surfaces of arbitrarily high genus. One algorithm is specifically presented to illustrate the adaption of the theory herein to the constructions in a work of H. C. Lyons [5]. Also displayed is a connection of some of the constructions available from this paper with Haken’s theory of normal surfaces. Every knot space of a non-trivial knot contains an incompressible surface of genus one which is parallel to the boundary of the knot space. So, incompressible surfaces of minimum genus are not always of interest as characteristic surfaces. It is suggested in this work that incompressible surfaces of some small, but not minimal, genus are good candidates for surfaces characteristic of the topology of a knot space. Interest here is in the structural properties of the topologies of knot spaces which provide and illustrate the necessity of considering characteristic surfaces of this kind.

Algorithms have been found in several instances to effectively determine incompressible surfaces in a given 3-manifold ([3],[6],[7]). The algorithms in this paper initially direct the construction of members of a class of 3-manifolds, then follow with prescriptions for a class of incompressible surfaces in each of these 3-manifolds.

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