# Winding number

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 *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 (*z*_{0}) 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 *z*_{0} is a point in **C** not on γ, then the *winding number* of γ with respect to *z*_{0} (alternately called the *index* of γ with respect to *z*_{0}) is defined by the formula:

This is verifiable from applying the Cauchy integral formula — the integral will be a multiple of 2πi, since each time γ goes about *z*_{0}, 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 *Q* ≠ *P*. 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 *m*σ.

## Generalizations

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