Winding number
From Free net encyclopedia
- The term winding number may also refer to the rotation number of an iterated map.
Image:Winding number example.png In mathematics, the winding number is a topological invariant playing a leading role in complex analysis.
Intuitively, the winding number of a curve γ with respect to a point z0 is the number of times γ goes around z0 in a counter-clockwise direction (number of turns).
In the image on the right, the winding number of the curve (C) about the inner point pictured (z0) is 3, since the curve makes three full revolutions around the point. The small loop on the left does not go around the point and so has no effect overall. Note that if the direction of the curve were reversed, the winding number would be −3 instead of 3.
One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated on the right has a winding number of 4 (or -4), because the small loop is counted.
Formal definitions
Formally, the winding number is defined as follows:
If γ is a closed rectifiable curve in C, and z0 is a point in C not on γ, then the winding number of γ with respect to z0 (alternately called the index of γ with respect to z0) is defined by the formula:
- <math> I(\gamma, z_0) = \frac{1}{2\pi i} \int_\gamma \frac{dz}{z - z_0} .</math>
This is verifiable from applying the Cauchy integral formula — the integral will be a multiple of 2πi, since each time γ goes about z0, we have effectively calculated the integral again.
The winding number is used in the residue theorem.
Another way of looking at this is that the complement of a point in the plane in homeomorphic 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 <math>\mathbf{S} \to \mathbf{S} : s \mapsto s^n</math>, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a topological space to the circle is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the integers Z and the winding number of a complex curve is just its homotopy class. In physics, winding numbers are frequently called topological quantum numbers.
Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index.