On Positively Turning Immersions

Abstract

Let $\gamma :{S^1} \to {\mathbf {C}}$ be a ${C^2}$ immersion of the circle. Let $k$ be the number of zeros of $\gamma$ and suppose $d\arg \gamma ({e^{i\theta }})/d\theta > 0$ for $\gamma ({e^{i\theta }}) \ne 0$; then $\operatorname {twn} \gamma = k/2 + {(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma$ where $\gamma$ is the tangent winding number, and $A = {S^1} - {\gamma ^{ - 1}}(0)$. 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 ${(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma$ represents a generalization of the notion of the winding number of $\gamma$ about $0$.