Braids and Signatures

In the plane R, we consider the sequence of points xi = (i, 0) (for i = 1, 2, . . .) and we denote by D(r) the disc of radius r, centered at the origin. In the space Xn of n-tuples of distinct points of D(n + 1/2), we consider the equivalence relation that identifies two n-tuples if one is obtained from the other by a permutation of the indices. We denote by X̃n the quotient space and by πn : Xn → X̃n the natural projection. The fundamental group of X̃n, based at πn(x 0 1, x 0 2, . . . , x 0 n), is called the n-th Artin braid group and is denoted by Bn; its elements are called braids. Any braid γ in Bn is represented by a path t ∈ [0, 1] 7→ (x1, x2, . . . , xn) ∈ Xn i.e. by a system of n disjoint arcs t 7→ (t, xi) in the cylinder [0, 1]×D2(n + 1/2), such that πn(x 1 1, x 1 2, . . . , x 1 n) = πn(x 0 1, x 0 2, . . . , x 0 n). The identification (x, 0) ≈ (x, 1) for all x inD(n+1/2) produces a finite collection of simple closed oriented curves in the solid torus R/Z × D(n + 1/2), images of the arcs t 7→ (t, xi). The usual embedding of the solid torus in 3-space R and the compactification of R with a point at infinity, allow us to associate with any braid γ an oriented link i.e. a collection of disjoint embeddings of an oriented circle in the 3-sphere S, called the closed braid associated with γ, and denoted by γ̂ (see Figure 1).