Torus Knots

Date of Award


Degree Type


Degree Name

Master of Science (MS)


Mathematics and Statistics

First Advisor

L. D. Loveland


We have compiled here many interesting results concerning a particular collection of knots called torus knots. Torus knots are merely simple closed curves imbedded in an unknotted torus T2 in E3. We show that the fundamental groups of T2, π(T2), is the direct product of the additive group of integers with itself. The ordered pair (p, q) in Z x Z determines an equivalence class of loops on the torus, and we show in Section II that the class [(p, q)] contains a loop whose image is a simple closed curve if and only if p and q are relatively prime. A torus knot in the loop class [(p, q)] is denoted Kp, q. It is natural to ask which of the knots Kp, q are equivalently imbedded in E3.

One means of answering this question is to observe the algebraic structures of the corresponding know groups π(E3 - Kp, q) and π(E3 - Kr,s) are not isomorphic, then it follows that E3 - Kp,q and E3 - Kr,s are not homeomorphic; consequently Kp,q and Kr,s are not equivalent knots. The definition and general properties of the fundamental group of a topological space are discussed in Section I of this report. In Section IV the fundamental group of E3 - Kp,q is shown to have the group presentation {a,b I ap = bq}. We will show that these groups are determined by the integers p and q, from which it follows that there are infinitely many non-equivalent torus knots.

Illustrations are use extensively to aid the reader, and an entire section is devoted to the development of an algorithm for picturing torus knots. This algorithm, Section III, provides us with an intuitive feeling for the significance of p and q in determining Kp,q. Finally, in Section V, a second knot type invariant, called the genus of the knot, is developed. The genus of a knot is a nonnegative integer assigned to the knot in a particular way. We will show that there exist torus knots of arbitrary genus and construct an infinite collection of knots, all having genus 1, none of which is a torus knot.

The material in this report comes from many sources. In many cases the proofs and illustrations were created by the author. We do not know of any similar compilation of facts relating to a specific class of knots and we hope that this report might be of use to other students of knot theory.

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