Self-intersections of curves on surfaces

Abstract

Let M be a compact orientable surface with nonempty boundary (χ(M) < 0) and fundamental group Γ. Let γ be a geodesic on M (with a fixed hyperbolic structure), and let W be a (cyclically reduced) word in a fixed set Γ̄ of generators of Γ which represents γ. In this paper, we give an algorithm to count the number of self-intersections of γ in terms of W, generalizing a result of Birman and Series, where an algorithm was given to decide if γ was simple. Some applications of the algorithm to surfaces with one boundary and the Markoff spectrum are also given. © 1996 Kluwer Academic Publishers.