MAKING CURVES MINIMALLY CROSSING BY REIDEMEISTER MOVES

Let C-1,..., C-k be a system of closed curves on a triangulizable surf ace S. The system is called minimally crossing if each curve C-l has a minimal number of self- intersections among all curios C-l'' freely h omotopic to C-l and if each pair C-i, C-j has a minimal number of inte rsections among all carve pairs C-i'', C-j'' freely homotopic to C-i, C-j respectively (i, j = 1 ,.... k, i not equal j). The system is call ed regular if each point traversed at least twice by these curves is t raversed exactly twice, and forms a crossing. We show that wt 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 incre ase. It implies a finite algorithm lu make a given system of curves mi nimally crossing by Reidemeister moves. (C) 1997 Academic Press.