= Winding number
{title2=$\operatorname{wind}(\gamma,w)$}
{wiki}
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> $\gamma$ avoiding $w$,
$$
\operatorname{wind}(\gamma,w)
=\frac1{2\pi i}\int_\gamma\frac{dz}{z-w}.
$$
Equivalently, continuously follow an <argument> of $\gamma(t)-w$ along one journey and divide its total change by $2\pi$. 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.<Topic 11: Argument Principle by Jeremy Orloff>[§11.2, definition of winding number and §11.2.1]{ref}
For $\gamma(t)=e^{it}$, $0\leq t\leq2\pi$, 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>.
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