Publication Date

5-2012

Advisor(s) - Committee Chair

Dr. Claus Ernst, Director, Dr. Uta Ziegler, Dr. John Spraker

Degree Program

Department of Mathematics and Computer Science

Degree Type

Master of Science

Abstract

Motivated by the action of XER site-specific recombinase on DNA, this thesis will study the topological properties of a type of local crossing change on oriented knots and links called nullification.

One can define a distance between types of knots and links based on the minimum number of nullification moves necessary to change one to the other. Nullification distances form a class of isotopy invariants for oriented knots and links which may help inform potential reaction pathways for enzyme action on DNA. The minimal number of nullification moves to reach a è-component unlink will be called the è-nullification number.

This thesis will demonstrate the relationship of the nullification numbers to a variety of knot invariants, and use these to solve the è-nullification numbers for prime knots up to 10 crossings for any è. A table of nullification numbers for oriented prime links up to 9 crossings is also presented, but not all cases are solved.

In addition, we examine the families of rational links and torus links for explicit results on nullification. Nullification numbers of torus knots and a subfamily of rational links are solved. In doing so, we obtain an expression for the four genus of said subfamily of rational links, and an expression for the nullity of any torus link.

Disciplines

Applied Mathematics

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