Winding number ( $wind (γ, w)$ )

The winding number records how many net turns a closed piecewise smooth curve makes around a chosen point

w

. Counterclockwise turns contribute positively, clockwise turns negatively. It measures a traversal, including repeated visits; a drawing of the curve's outline alone can lose that information. For a closed curve

γ

avoiding

w

wind (γ, w) = \frac{1}{2 πi} \int_{γ} \frac{d z}{z - w} .

(1)

Equivalently, continuously follow an argument of

γ (t) - w

along one journey and divide its total change by

2 π

. The start and endpoint agree as complex numbers, so the change is an integer number of full turns. The principal argument at the endpoints cannot retain those turns; a continuous choice along the parametrized path can.^[ref]

For

γ (t) = e^{i t}

0 \leq t \leq 2 π

, the winding number about zero is one. Traversing the same unit circle twice makes it two; reversing its orientation of a contour makes it negative. The winding number about

w = 2

is zero: the same journey now stays to one side of the point being measured. Joining one counterclockwise loop and one clockwise loop around zero gives net winding number zero even though both loops encircle zero.

The winding number does not change during a homotopy of closed curves avoiding

w

. This is why it is useful: complicated curves can often be deformed to simple ones whose count is obvious. The argument principle transfers that robust geometric integer to a count of zeros of a function and poles. The example in boundary sampling can miss winding shows why finite samples of a curve need control between samples before they can certify the same winding number.

Table of contents
- Winding number of a continuous closed path Winding number
  - Continuous logarithm lifting criterion Winding number of a continuous closed path

Winding number of a continuous closed path

 0  0

γ : [0, 1] \to C ∖ {0}

is continuous and closed, choose a continuous lift

θ

such that

\frac{γ ( t )}{∣ γ ( t ) ∣} = e^{2 πi θ (t)} .

(1)

Then

wind (γ, 0) = θ (1) - θ (0) \in Z

. This agrees with the contour-integral definition when the path is piecewise smooth.

Continuous logarithm lifting criterion ( $f = e^{g} ⟺ wind (f \circ γ, 0) = 0 for every loop γ$ )

 0  0

On a locally path-connected space, on each path component choose a value of a complex logarithm of

f

at a base point. Continue it along paths using local logarithms. Independence of the path is exactly vanishing of the winding number of a continuous closed path for all loops. This constructs a continuous function

g

with

f = e^{g}

. The implication from a logarithm to zero winding requires no local connectivity. For applications to arbitrary planar compact sets, apply the criterion on polygonal neighborhoods rather than assuming the compact set is locally path-connected.