Loops around a hole

The number of times a closed path goes around a point is its winding number. If the point is $a$ and the path is traced by a complex number $z(t)$, the winding number is the total change in the argument of $z - a$, divided by $2\pi$. It is always a whole number, because the path ends where it began.

Here is the key fact: if you deform a loop continuously without ever passing through $a$, its winding number cannot jump — a whole number that changes continuously cannot change at all. So loops in the plane with a hole are sorted by their winding number, and two loops can be deformed into each other exactly when their winding numbers agree. In the language of topology, the fundamental group of the plane with one point removed is $\mathbb Z$.

Complex analysis computes the same number with an integral:

$$\frac{1}{2\pi i}\oint \frac{dz}{z - a} = \text{winding number}.$$

Winding numbers even give a proof sketch of the fundamental theorem of algebra. A polynomial $z^n + \dots$ maps a very large circle to a loop that winds $n$ times around $0$, while a tiny circle near the origin maps to a loop that winds $0$ times around $0$ unless $p(0) = 0$. Shrinking the circle changes the winding number from $n$ to $0$, which can only happen if the image loop passes through $0$ somewhere — that is, if the polynomial has a root. (Turning this sketch into a proof takes care; see the fundamental-group lesson.)

With two holes the story changes: loops are no longer classified by two winding numbers, because going around one hole and then the other is different from the reverse order. The fundamental group becomes non-commutative.