Making Curves Minimally Crossing by Reidemeister Moves

LetC1, …, Ckbe a system of closed curves on a triangulizable surfaceS. The system is calledminimally crossingif each curveCihas a minimal number of self-intersections among all curvesC′ifreely homotopic toCiand if each pairCi,Cjhas a minimal number of intersections among all curve pairsC′i, C′jfreely homotopic toCi, Cjrespectively (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.