Winding number

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The term winding number may also refer to the rotation number of an iterated map.

A point z₀ and a curve C

In mathematics, the winding number is a property of a curve, which plays a leading role in complex analysis and algebraic topology. It is the fundamental case of degree of a continuous mapping.

1 Intuitive description
2 Turning number
3 Formal definitions
- 3.1 Complex analysis
- 3.2 Topology
4 See also
5 External links

[edit] Intuitive description

Intuitively, the winding number of a curve γ with respect to a point $z 0$ is the number of times γ goes around $z 0$ 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.

[edit] Turning number

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.

This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss map.

This is called the turning number.

[edit] Formal definitions

There are distinct but related concepts of a winding number in complex analysis and in topology. It may be defined as follows.

[edit] Complex analysis

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:

$I(\gamma, z_0) = \frac{1}{2\pi i} \int_\gamma \frac{dz}{z - z_0} .$

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.

[edit] Topology

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 $S^1 \to S^1 : s \mapsto s^n$ , where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a 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.