Knots

Abstract

A knot is a subset of ℝ³ which is homeomorphic to a circle. We restrict ourselves here to tame knots, i.e. those which are simple closed polygons in ℝ³. A link with k components is a subset of ℝ³ which consists of k disjoint knots (components of the link). A link is oriented if each component is prescribed one of the two possible directions of traversal. Since the beginning of the knot theory, knots are regularly projected onto the plane and thus represented by a (planar knot) diagram. The notion of the knot or link diagram is very intuitive. Some examples of oriented diagrams appear in Figure 8.1. A link diagram is simply a 4-regular topological planar graph where the vertices are replaced by the crossings. At each crossing it is specified which transition goes up and which one goes down. For oriented links this leads to two kinds of crossings, denoted by the signs + and −. These signs are defined by Figure 8.2. Two knots are equivalent if they can be transformed into each other by a continuous deformation of the ambient space. The equivalence may be captured combinatorially by the Δ- move in ℝ³ and its reverse move. If K is a knot in ℝ³, i.e. a closed simple polygon by our assumption, then a Δ-move consists in replacing a straight line segment l of K by the other two sides of a triangle T having sides l, k, j. It is assumed that T and K intersect only in l.