Rotation number of an immersed oriented circle in the plane

Let $S^1$ be oriented and parametrised as $\mathbb{R}\mathrm{/}2\pi \mathbb{Z}$ with the positive orientation, let ${\mathrm{\partial }}_{\theta }$ be the corresponding positively oriented unit tangent vector field of $S^1$, and let $f:S^1\to {\mathbb{R}}^2$ be a smooth immersion, that is, a regular closed curve (Immersions, submersions, and constant-rank maps, Smooth manifolds and their smooth charts). Writing $T{\mathbb{R}}^2\cong {\mathbb{R}}^2\times {\mathbb{R}}^2$ for the canonical identification, the differential (The differential of a smooth map, The tangent bundle as a disjoint union) gives for each $\theta$ the velocity vector $d{f}_{\theta }({\mathrm{\partial }}_{\theta })$, which is nonzero because $f$ is an immersion.

The normalised velocity, or unit tangent, of $f$ is the map ${\tau }_f:S^1⟶S^1,{\tau }_f(\theta )=\frac{d{f}_{\theta }({\mathrm{\partial }}_{\theta })}{\mid d{f}_{\theta }({\mathrm{\partial }}_{\theta })\mid },$ which is continuous because the velocity never vanishes and every nonzero vector is a positive multiple of a unique unit vector.

The rotation number (also rotation index) of $f$ is $\mathrm{rot}(f):=\mathrm{deg}({\tau }_f)\in \mathbb{Z},$ the degree of the unit tangent map. Concretely, after choosing the base point ${\theta }_0=[0]$, identifying the unit circle with $\mathbb{R}\mathrm{/}\mathbb{Z}$ by the standard parametrisation and rotating the target circle by a constant so that ${\tau }_f({\theta }_0)$ corresponds to $[0]$, the class of ${\tau }_f$ is a based loop in the sense of The degree of a based circle loop, and its degree is the integer $\mathrm{rot}(f)$; the constant rotation of the target plane changes neither the winding number nor the degree, and the degree descends to based loop classes (Degree defines a function $\mathrm{Deg}:{\pi }_1(S^1,[0])\to \mathbb{Z}$), so the value is independent of the choice of ${\theta }_0$ and of the lift used to compute it (Two based circle loops are path-homotopic if and only if they have equal degree).

Equivalent formulations, all taking the same value:

• The velocity curve $t\mapsto f^{\mathrm{\prime }}(t)$ is a closed $C^1$, hence rectifiable, loop in ${\mathbb{C}}^{\times }={\mathbb{R}}^2\setminus \{0\}$, and $\mathrm{rot}(f)$ is its winding number about the origin: the normalised velocity $t\mapsto f^{\mathrm{\prime }}(t)\mathrm{/}\mid f^{\mathrm{\prime }}(t)\mid$ is exactly ${\tau }_f$, and after the constant nonzero complex multiplication $\gamma \mapsto \gamma \mathrm{/}\gamma (t_0)$ of the target plane, which changes neither the winding number about $0$ nor the degree of the normalised loop, For loops in C times, the winding number about 0 equals the circle degree identifies $n(\gamma ,0)$ with the degree of the normalised loop, while Winding number identifies the fundamental group of C times with the integers identifies that winding number with the class in ${\pi }_1({\mathbb{C}}^{\times },1)\cong \mathbb{Z}$.
• $\mathrm{rot}(f)$ equals $\frac{1}{2\pi }$ times the total signed turning angle of the tangent, i.e. the rotation index of the regular closed plane curve $f$ in the sense of Rotation index of a regular closed plane curve, where the tangent angle is lifted continuously and the corner jumps are zero for a smooth curve.

The normalisation is fixed by the round unit circle: the immersion $\theta \mapsto (\cos \theta ,\sin \theta )$ traversed once in the positive direction has unit tangent $(-\sin \theta ,\cos \theta )$ winding once positively, hence rotation number $+1$, and its reverse has rotation number $-1$. Reversing the orientation of the domain negates the rotation number, while a regular (orientation-preserving) reparametrisation leaves it unchanged. Indeed, if $h$ is a positively oriented circle diffeomorphism with increasing lift $H$ satisfying $H(\theta +2\pi )=H(\theta )+2\pi$, then the chain rule gives ${\tau }_{f\circ h}={\tau }_f\circ h$: composing a tangent-angle lift with $H$ preserves its total increment. For the reversal $a(\theta )=-\theta$, ${\tau }_{f\circ a}(\theta )=-{\tau }_f(-\theta )$, so an angle lift is $\alpha (-\theta )+\pi$ when $\alpha$ lifts ${\tau }_f$; its total increment is the negative of that of $\alpha$.

No convexity, simplicity, self-intersection restriction or properness is imposed; $\mathrm{rot}$ is an invariant of the oriented immersed circle, not of its image.