Winding Number, Definition

Definition of Winding Number

Take a moment to review the winding number of a loop in the plane. Here the origin is fixed, and different curves are considered. Each curve is projected onto the unit circle, whereupon the winding number becomes the degree. Homotopic curves produce homotopic projections in the unit circle, giving maps of the same degree. Therefore homotopic curves have the same winding number. So far so good.

In this section we are going to turn things around. The curve is fixed in the plane, and various points, not on the curve, will act as the origin.

Let c be a closed curve in the plane, and let u be a point not on the curve. treat u as the origin, and convert each c(x) into polar coordinates. Now θ(c(x)) is a continuous map from the circle into the circle, and its degree is the winding number of u.

Homotopic

If c and d are two curves that are homotopic, and the homotopy never touches u, then the homotopy can be projected onto θ, whence the induced maps from S¹ into S¹ are also homotopic. Therefore u has the same winding number, relative to c and d.

Path Components

Let u and v be path connected via the path p, such that p never touches the curve c. Let the path become a homotopy that slides u over to v. This induces a homotopy on θ(c(x)). (If you like analysis, you can verify two dimensional continuity, as required by a proper homotopy.) Therefore the winding number of u equals the winding number of v. In fact, the winding number is specific to each path component in the plane, once c is removed.

Since the curve is bounded, select u outside this bound. Now the curve can shrink linearly to a point, without touching u. The winding number of u, relative to a point, is 0, hence the winding number of u, relative to c, is also 0. The infinite region of the plane that lies "outside" of c has winding number 0.

Let's illustrate with the circle, i.e. c(x) maps the circle onto itself. The inside is a path component, and so is the outside. As shown above, the outside has winding number 0. Let u be the origin, and the curve has degree 1, whence the inside has winding number 1. Since these numbers are different, the inside and outside are distinct path components. The circle cuts the plane into two pieces, the inside and the outside.

It is possible, however, for separate components to have the same winding number. Let c be a figure 8, where both the top and bottom circles run counterclockwise. When u is inside the top circle, the bottom circle can shrink down to a poine, giving a degree of 1. Similarly, the inside of the bottom circle has degree 1; yet this represents a separate path component.