Winding number

The winding number of a closed curve around a point is an integer that counts its net turns around that point, with direction determining the sign.

Argument principle: A theorem relating zeros and poles of a meromorphic function inside a contour to the contour integral of its logarithmic derivative. The theorem identifies this integer with the number of zeros minus poles enclosed by a complex contour.
Closed curve: A continuous path whose starting point and endpoint coincide. The winding number applies to loops rather than open paths.
Rouché's theorem: A complex analysis theorem that compares the numbers of zeros of two functions inside a contour. It uses equality of boundary winding numbers to establish equal zero counts.
Linking number: An integer invariant measuring how two disjoint closed curves are linked in three-dimensional space. It counts linking between two curves, rather than turns of one curve around a point.
Topological degree: An integer invariant that counts, with orientation, how a map covers a target space. Winding number is the degree of a map from a circle to a circle.
Orientation (mathematics): A consistent choice of direction or handedness on a mathematical space or object. Reversing the curve’s orientation reverses the sign of its winding number.
Poincaré–Hopf theorem: A theorem equating the sum of vector-field indices on a compact manifold with its Euler characteristic. Indices of planar vector-field zeros are local winding numbers.
Rotation number: An invariant measuring average angular advance under iterations of a circle homeomorphism. It also records turns, but for repeated dynamics rather than a single closed curve.
Complex logarithm: A multivalued inverse of the complex exponential, whose values differ by integer multiples of 2πi. A continuous logarithm along a closed curve accumulates 2πi times its winding number.
Fundamental group: A group encoding the homotopy classes of loops based at a point in a space. Loops around a puncture are classified by their integer winding counts.

Linked from 14 pages