The winding number of a contour about a point
, denoted
, is defined by
and gives the number of times 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 compact, oriented manifolds
of the same dimension. In the language of the 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 degree of the map