On Positively Turning Immersions

Abstract

Abstract.Let y: S1 -» C be a C2 immersion of the circle. Let k be the number of zeros of y and suppose da.tg,y(e )/d9 > 0 for y(e'e) # 0; then twny = k/2 + (2ir)~ Sa daxgy where twny is the tangent winding number, and A = S1 -y~'(0). This generalizes the theorem of Cohn that if p is a self-inversive polynomial, the number of zeros of p' in \z\ > 1 is the same as the number of zeros of p in \z\ > 1. For k = 0, this is a topological generalization of Lucas' theorem. We show how (2ir) fA rfargy represents a generalization of the notion of the winding number of y about 0.