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- The term winding number may also refer to the rotation number of an iterated map.

In mathematics, the winding number of closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.

Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics.

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 rotations that the object makes around the origin.

When counting the total number of rotations, 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 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. The following pictures show curves with winding numbers between −2 and 3:

$cdots$ | ||||

−2 | −1 | 0 | ||

$cdots$ | ||||

1 | 2 | 3 |

- $x\; =\; x(t)quadtext\{and\}quad\; y=y(t)qquadtext\{for\; \}0\; leq\; t\; leq\; 1.$

If we think of the parameter t as time, then these equations specify the motion of an object in the plane between and . The path of this motion is a curve as long as the functions x(t) and y(t) are continuous. This curve is closed as long as the position of the object is the same at and .

We can define the winding number of such a curve using the polar coordinate system. Assuming the curve does not pass through the origin, we can rewrite the parametric equations in polar form:

- $r\; =\; r(t)quadtext\{and\}quad\; theta\; =\; theta(t)qquadtext\{for\; \}0\; leq\; t\; leq\; 1.$

The functions r(t) and θ(t) are required to be continuous, with . Because the initial and final positions are the same, θ(0) and θ(1) must differ by an integer multiple of 2π. This integer is the winding number:

- $text\{winding\; number\}\; =\; frac\{theta(1)\; -\; theta(0)\}\{2pi\}$

This defines the winding number of a curve around the origin in the xy plane. By translating the coordinate system, we can extend this definition to include winding numbers around any point p.

- $dtheta\; =\; frac\{1\}\{r^2\}\; left(x,dy\; -\; y,dx\; right)quadtext\{where\; \}r^2\; =\; x^2\; +\; y^2.$

- $text\{winding\; number\}\; =\; frac\{1\}\{2pi\}\; oint\_C\; ,frac\{x\}\{r^2\},dy\; -\; frac\{y\}\{r^2\}$

- $dz\; =\; e^\{itheta\}\; dr\; +\; ire^\{itheta\}\; dtheta!,$

and therefore

- $frac\{dz\}\{z\}\; ;=;\; frac\{dr\}\{r\}\; +\; i,dtheta\; ;=;\; d[ln\; r\; ]\; +\; i,dtheta.$

The total change in ln(r) is zero, and thus the integral of dz ⁄ z is equal to i multiplied by the total change in θ. Therefore:

- $text\{winding\; number\}\; =\; frac\{1\}\{2pi\; i\}\; oint\_C\; frac\{dz\}\{z\}.$

More generally, the winding number of C around any complex number a is given by

- $frac\{1\}\{2pi\; i\}\; oint\_C\; frac\{dz\}\{z\; -\; a\}.$

This is a special case of the famous Cauchy integral formula. Winding numbers play a very important role throughout complex analysis (c.f. the statement of the residue theorem).

The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is homotopy equivalent to the circle, such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps $S^1\; to\; S^1\; :\; s\; mapsto\; s^n$, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a circle to a topological space is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the integers Z and the winding number of a complex curve is just its homotopy class.

Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index.

This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss map.

This is called the turning number.

- Residue theorem
- Argument principle
- Linking coefficient
- Topological quantum number
- Topological degree theory

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Last updated on Friday September 26, 2008 at 12:11:57 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 26, 2008 at 12:11:57 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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