SELF-INTERSECTION NUMBERS OF PATHS IN COMPACT SURFACES

Abstract

In this paper, we give an algorithm to calculate the minimal self-intersection number of paths in a compact surface with boundary representing a given element of the free group F(x1, x2, …, xn). In particular, this algorithm says whether or not a word in x1, x2, …, xn is representable by a simple path. Our algorithm is simpler than similar algorithms given previously. In the case of a disk with n holes the problem is equivalent to the problem of deciding which relators can appear in an Artin n-presentation.