Knot tabulations and related topics

Summary

Introduction

The aim of this article is to examine some of the ideas connected with the problem of classifying 1-dimensional knots. As the title suggests, the flavour is intended to be rather pragmatic. We follow through the history of Knot tabulations, from the pre-topological dark ages of the last century up to the present day.

Dowker developed an interest in knot tabulations in the 1960's, and collaborated recently with the present author in the classification of knots of up to 13 crossings.

It is hoped that the expository parts of sections 1, 3 and 4 will make the article reasonably self-contained. We pay only cursory attention to “abelian” knot theory, as this is dealt with extensively in Cameron Gordon's survey article (Gor).

I would like to thank John Conway, Raymond Lickorish and Larry Siebenmann for valuable conversations.

Preliminaries

A knot is a submanifold of S³ homeomorphic to S¹. Below are drawings of three famous Knots.

Two Knots K, L are equivalent if there exists an autohomeomorphism of S³ mapping K onto L. The equivalence class of K is its knot type. We shall consider exclusively Knot types with piecewise linear (or, equivalently, smooth) representatives, in order to steer clear of the virtually unexplored world of wild knots, i.e. Knot types with no piecewise linear representative. The reader interested in wild Knots can find information and references in (Fox₃). We shall often use the word “Knot” as an abbreviation for “Knot type”, when there is no danger of confusion.