On an invariant of pure braids

Abstract

Consider a pure braid β. It can be expressed as n disjoint strandswhere X(0) = X(1). We assume β to be generic. This means that there are finite number of values. . , m, neither four points of X(t) lie on one circle or line, and the set X(t k ), k = 1, . . . , m contains exactly one quadruple of points which lie on one circle (or line). Then for any regular value t = t k , the set X(t) determines a unique Delaunay triangulation G(t).Fix an integer r ≥ 3. Denote q = e iπ r ∈ C. For any regular value t, consider the set F r (t) = F r (G(t)) of admissible colourings of the edges of the Delaunay triangulation G(t), i.e. maps f : E(G(t)) → N ∪ {0} such that for any triangle of G(t) with edges a, b, c one has