Winding number

File:Winding number example.png
A point z0 and a curve C

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 tangent to the curve, or equivalently, if it is a path followed in time, the winding number about the origin of the velocity. In the example we get 4 or -4, now the small loop is counted.

Formal definitions

Formally, the winding number is defined as follows:

If γ is a closed 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:

${\displaystyle I(\gamma ,z_{0})={\frac {1}{2\pi i}}\int _{\gamma }{\frac {dz}{z-z_{0}}}.}$

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.

In more abstract terms, the fundamental group of the complement of a point P in the plane is infinite cyclic. Choose a generator σ in the positively-oriented direction, of the fundamental group with base point some fixed point QP. Create a loop based at Q from C, by joining Q to C by an arc to the starting point of C, going round c, then going back the same way to Q. The winding number will be m if the class of this loop in the fundamental group is mσ.

Generalizations

In physics, a winding number is frequently used as a synonym for the topological quantum number of a topological soliton.