In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the winding number or winding index of a closed
curve in the
plane around a given
point is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's
number of turns. The winding number depends on the
orientation of the curve, and it is
negative if the curve travels around the point clockwise.
Winding numbers are fundamental objects of study in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, and they play an important role in
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
,
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
,
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originate ...
,
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
(such as in
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
).
Intuitive description
Suppose we are given a closed, oriented curve in the ''xy'' plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin.
When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative. For example, if the object first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three.
Using this scheme, a curve that does not travel around the origin at all has winding number zero, while a curve that travels clockwise around the origin has negative winding number. Therefore, the winding number of a curve may be any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. The following pictures show curves with winding numbers between −2 and 3:
Formal definition
Let
be a continuous closed path on the plane minus one point. The winding number of
around
is the integer
:
where
is the path written in polar coordinates, i.e. the lifted path through the
covering map
:
The winding number is well defined because of the
existence and uniqueness of the lifted path (given the starting point in the covering space) and because all the fibers of
are of the form
(so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed.
Alternative definitions
Winding number is often defined in different ways in various parts of mathematics. All of the definitions below are equivalent to the one given above:
Alexander numbering
A simple
combinatorial rule for defining the winding number was proposed by
August Ferdinand Möbius in 1865
and again independently by
James Waddell Alexander II
James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ...
in 1928.
Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal. The winding number around (any point in) the unbounded region is zero. Finally, the winding numbers for any two adjacent regions differ by exactly 1; the region with the larger winding number appears on the left side of the curve (with respect to motion down the curve).
Differential geometry
In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, parametric equations are usually assumed to be
differentiable (or at least piecewise differentiable). In this case, the polar coordinate ''θ'' is related to the rectangular coordinates ''x'' and ''y'' by the equation:
:
Which is found by differentiating the following definition for θ:
:
By the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
, the total change in ''θ'' is equal to the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of ''dθ''. We can therefore express the winding number of a differentiable curve as a
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
:
:
The
one-form ''dθ'' (defined on the complement of the origin) is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
but not exact, and it generates the first
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
group of the
punctured plane. In particular, if ''ω'' is any closed differentiable one-form defined on the complement of the origin, then the integral of ''ω'' along closed loops gives a multiple of the winding number.
Complex analysis
Winding numbers play a very important role throughout complex analysis (c.f. the statement of the
residue theorem). In the context of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the winding number of a
closed curve in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
can be expressed in terms of the complex coordinate . Specifically, if we write ''z'' = ''re''
''iθ'', then
:
and therefore
:
As
is a closed curve, the total change in
is zero, and thus the integral of
is equal to
multiplied by the total change in
. Therefore, the winding number of closed path
about the origin is given by the expression
:
More generally, if
is a closed curve parameterized by