Elements of Classical Knot Theory

Abstract

The aim of this article is to present an elementary introduction to classical knot theory. The word classical means two things. First, it means the study of knots in the usual 3D space R 3 or S 3. It also designates knot theory before 1984. In section 1 we describe the basic facts: curves in 3D space, isotopies, knots, links and knot types. We then proceed to knot diagrams and braids. Finally we introduce the useful notion of tangle due to John Conway. In section 2 we present some important problems of knot theory: classification, chirality, search for and computation of invariants. In section 3 we give a brief description of some knot families: alternating knots, two-bridge knots, torus knots. Within each family, the classification problem is solved. In section 4 we indicate two ways to introduce some structure in knot types: via ideal knots and via the knot complement.