Making Curves Minimally Crossing by Reidemeister Moves

Let C1 , ..., Ck be a system of closed curves on a triangulizable surface S. The system is called minimally crossing if each curve Ci has a minimal number of selfintersections among all curves C$i freely homotopic to Ci and if each pair Ci , Cj has a minimal number of intersections among all curve pairs C$i , C$j freely homotopic to Ci , Cj respectively (i, j=1, ..., k, i{ j). The system is called regular if each point traversed at least twice by these curves is traversed exactly twice, and forms a crossing. We show that we can make any regular system minimally crossing by applying Reidemeister moves in such a way that at each move the number of crossings does not increase. It implies a finite algorithm to make a given system of curves minimally crossing by Reidemeister moves. 1997 Academic Press