Publication Date

5-2010

Advisor(s) - Committee Chair

Dr. Uta Ziegler (Director), Dr. Claus Ernst, Dr. Mustafa Atici

Degree Program

Department of Mathematics and Computer Science

Degree Type

Master of Science

Abstract

The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(n) time where n is the number of crossings in the knot, and it derives a precise representation of the knot’s nice drawing in O(n) time (The rendering of the drawing is not O(n).).
Central to the algorithm is a special type of rooted binary tree which represents a distinct prime, alternating Conway algebraic knot. Each leaf in the tree represents a crossing in the knot. The algorithm first generates the tree and then modifies such a tree repeatedly to reduce the number of its leaves while ensuring that the knot type associated with the tree is not modified. The result of the algorithm is a tree (for the knot) with a minimum number of leaves. This minimum tree is the basis of deriving a 4-regular plane map which represents the knot embedding and to finally draw the knot’s diagram.

Disciplines

Discrete Mathematics and Combinatorics | Mathematics | Numerical Analysis and Computation