Intuition Behind Winding Numbers

On intuitive way to think of the winding number is to imagine the curve made of a very slippery elastic cord, and to think of the point as being replaced by a post or spool cemented to the plane. When the cord is released, it will whip around a bit, but eventually it will shrink down tightly onto the spool. The cord will be wrapped around the spool some number of times, and this is basically the winding number.

The mathematical description of the winding number also distinguishes the direction of the wrapping. If we think of our cord as having tiny arrows printed on it, it will either wrap clockwise or counter-clockwise around the spool. If we agree to count turns counter-clockwise positively and clockwise ones negatively, we then have the mathematical winding number.

The winding number plays an important role in the mathematical area called complex analysis. Complex analysis studies functions from the complex plane to itself. As a set, the complex plane is just $R^2$, the Euclidean plane. However, when we think of it as the complex plane, each point corresponds to a complex number, with the real part being the $x$-coordinate, and the imaginary part being the $y$-coordinate.

The standard arithmetic operations for complex numbers has a very geometric flavor, involving lengths and rotations. This imposes strict geometric conditions on the behavior of functions of complex numbers.

In science, many phenomena such as heat conduction and fluid flow are closely related to functions of complex numbers called holomorphic functions. Because of all the geometric structure, in a very deep and subtle way information about the global properties of these functions on an entire region can be computed from information about the function on a curve or at some special points.

The winding number formula

<IMG WIDTH="191" HEIGHT="62" ALIGN="MIDDLE" BORDER="0" SRC="img4.gif" ALT="$\displaystyle I(\gamma,p) = \frac{1}{2 \pi i} \int_{\gamma} \frac{dz}{z - p} $">

is a good example of this phenomenon. Without going into detail, the integral on the right hand side adds up values of the function

<IMG WIDTH="52" HEIGHT="60" ALIGN="MIDDLE" BORDER="0" SRC="img5.gif" ALT="$\displaystyle \frac{1}{z - p} $">

along the curve, and presto-chango out comes the winding number, which is a global property of the curve and $p$, and which seemingly has nothing to do with the values of some obscure function along the curve.

Amazingly, the value of this integral doesn't depend very much at all on the path we integrate around. If you believe this really is a formula for the winding number, the value shouldn't change unless we change the winding number. But as long as we avoid the point $p$ as we deform the curve any way we want, and the winding number won't change.