On positively turning immersions

Abstract

Let γ : S 1 → C \gamma :{S^1} \to {\mathbf {C}} be a C 2 {C^2} immersion of the circle. Let k k be the number of zeros of γ \gamma and suppose d arg ⁡ γ ( e i θ ) / d θ > 0 d\arg \gamma ({e^{i\theta }})/d\theta > 0 for γ ( e i θ ) ≠ 0 \gamma ({e^{i\theta }}) \ne 0 ; then twn ⁡ γ = k / 2 + ( 2 π ) − 1 ∫ A d arg ⁡ γ \operatorname {twn} \gamma = k/2 + {(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma where γ \gamma is the tangent winding number, and A = S 1 − γ − 1 ( 0 ) A = {S^1} - {\gamma ^{ - 1}}(0) . This generalizes the theorem of Cohn that if p p is a self-inversive polynomial, the number of zeros of p ′ p’ in | z | > 1 |z| > 1 is the same as the number of zeros of p p in | z | > 1 |z| > 1 . For k = 0 k = 0 , this is a topological generalization of Lucas’ theorem. We show how ( 2 π ) − 1 ∫ A d arg ⁡ γ {(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma represents a generalization of the notion of the winding number of γ \gamma about 0 0 .