Contour Winding Number

The contour winding number of a contour about a point , denoted , is defined by

and gives the number of times the curve passes (counterclockwise) around a point. Counterclockwise winding is assigned a positive winding number, while clockwise winding is assigned a negative winding number. The winding number is also called the index, and denoted .

The contour winding number was part of the inspiration for the idea of the Brouwer degree between two oriented compact manifolds of the same dimension. In the language of the Brouwer degree of a map, if is a closed curve (i.e., ), then it can be considered as a function from to . In that context, the winding number of around a point in is given by the Brouwer degree of the map

from the circle to the circle.