## Abstract

A *knot* is a subset of ℝ^{3} which is homeomorphic to a circle. We restrict ourselves here to *tame knots*, i.e. those which are simple closed polygons in ℝ^{3}. A *link* with *k* components is a subset of ℝ^{3} 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 ℝ^{3} and its reverse move. If *K* is a knot in ℝ^{3}, 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*.

## Keywords

Braid Group Weight System Jones Polynomial Vertex Model Reidemeister Move## Preview

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