Winding number

A simple combinatorial rule for defining the winding number was proposed by August Ferdinand Möbius in 1865[1] and again independently by James Waddell Alexander II in 1928.[2] Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal. The winding number around (any point in) the unbounded region is zero. Finally, the winding numbers for any two adjacent regions differ by exactly 1; the region with the larger winding number appears on the left side of the curve (with respect to motion down the curve).

In differential geometry, parametric equations are usually assumed to be differentiable (or at least piecewise differentiable). In this case, the polar coordinate θ is related to the rectangular coordinates x and y by the equation:

${\displaystyle d\theta ={\frac {1}{r^{2}}}\left(x\,dy-y\,dx\right)\quad {\text{where }}r^{2}=x^{2}+y^{2}.}$

Which is found by differentiating the following definition for θ:

${\displaystyle \theta (t)=\arctan {\bigg (}{\frac {y(t)}{x(t)}}{\bigg )}}$

By the fundamental theorem of calculus, the total change in θ is equal to the integral of dθ. We can therefore express the winding number of a differentiable curve as a line integral:

${\displaystyle {\text{winding number}}={\frac {1}{2\pi }}\oint _{C}\,\left({\frac {x}{r^{2}}}\,dy-{\frac {y}{r^{2}}}\,dx\right).}$

The one-form dθ (defined on the complement of the origin) is closed but not exact, and it generates the first de Rham cohomology group of the punctured plane. In particular, if ω is any closed differentiable one-form defined on the complement of the origin, then the integral of ω along closed loops gives a multiple of the winding number.

Winding numbers play a very important role throughout complex analysis (c.f. the statement of the residue theorem). In the context of complex analysis, the winding number of a closed curve ${\displaystyle \gamma }$ in the complex plane can be expressed in terms of the complex coordinate z = x + iy. Specifically, if we write z = re^iθ, then

${\displaystyle dz=e^{i\theta }dr+ire^{i\theta }d\theta \!\,}$

and therefore

${\displaystyle {\frac {dz}{z}}\;=\;{\frac {dr}{r}}+i\,d\theta \;=\;d[\ln r]+i\,d\theta .}$

As ${\displaystyle \gamma }$ is a closed curve, the total change in ln(r) is zero, and thus the integral of ${\displaystyle {\frac {dz}{z}}}$ is equal to ${\displaystyle i}$ multiplied by the total change in ${\displaystyle \theta }$. Therefore, the winding number of closed path ${\displaystyle \gamma }$ about the origin is given by the expression[3]

${\displaystyle {\frac {1}{2\pi i}}\oint _{\gamma }{\frac {dz}{z}}}$.

More generally, if ${\displaystyle \gamma }$ is a closed curve parameterized by ${\displaystyle t\in [\alpha ,\beta ]}$, the winding number of ${\displaystyle \gamma }$ about ${\displaystyle z_{0}}$, also known as the index of ${\displaystyle z_{0}}$ with respect to ${\displaystyle \gamma }$, is defined for complex ${\displaystyle z_{0}\notin \gamma ([\alpha ,\beta ])}$ as[4]

${\displaystyle \mathrm {Ind} _{\gamma }(z_{0})={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z_{0}}}={\frac {1}{2\pi i}}\int _{\alpha }^{\beta }{\frac {\gamma '(t)}{\gamma (t)-z_{0}}}dt}$.

This is a special case of the famous Cauchy integral formula.

Some of the basic properties of the winding number in the complex plane are given by the following theorem:[5]

Theorem. Let ${\displaystyle \gamma :[\alpha ,\beta ]\to \mathbb {C} }$ be a closed path and let ${\displaystyle \Omega }$ be the set complement of the image of ${\displaystyle \gamma }$, that is, ${\displaystyle \Omega :=\mathbb {C} \setminus \gamma ([\alpha ,\beta ])}$. Then the index of ${\displaystyle z}$ with respect to ${\displaystyle \gamma }$,

${\displaystyle \mathrm {Ind} _{\gamma }:\Omega \to \mathbb {C} ,\ \ z\mapsto {\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z}}}$,

is (i) integer-valued, i.e., ${\displaystyle \mathrm {Ind} _{\gamma }(z)\in \mathbb {Z} }$ for all ${\displaystyle z\in \Omega }$; (ii) constant over each component (i.e., maximal connected subset) of ${\displaystyle \Omega }$; and (iii) zero if ${\displaystyle z}$ is in the unbounded component of ${\displaystyle \Omega }$.

As an immediate corollary, this theorem gives the winding number of a circular path ${\displaystyle \gamma }$ about a point ${\displaystyle z}$. As expected, the winding number counts the number of (counterclockwise) loops ${\displaystyle \gamma }$ makes around ${\displaystyle z}$:

Corollary. If ${\displaystyle \gamma }$ is the path defined by ${\displaystyle \gamma (t)=a+re^{int},\ \ 0\leq t\leq 2\pi ,\ \ n\in \mathbb {Z} }$, then ${\displaystyle \mathrm {Ind} _{\gamma }(z)={\begin{cases}n,&|z-a|<r;\\0,&|z-a|>r.\end{cases}}}$

In topology, the winding number is an alternate term for the degree of a continuous mapping. In physics, winding numbers are frequently called topological quantum numbers. In both cases, the same concept applies.

The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is homotopy equivalent to the circle, such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps ${\displaystyle S^{1}\to S^{1}:s\mapsto s^{n}}$, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a circle to a topological space form a group, which is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the group of the integers, Z; and the winding number of a complex curve is just its homotopy class.

Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index.

The boundary of the regular Enneagram {9/4} winds around its centre 4 times, so it has a density of 4.

In polygons, the winding number is referred to as the polygon density. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem. By contrast, for a regular star polygon {p/q}, the density is q.