Difference between revisions of "Whitney-Graustein theorem"
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| + | The [[Winding number|winding number]] $W ( C )$ of a continuous closed curve $C$ in the plane, missing the origin, counts the algebraic number of times $C$ goes around the origin in the counterclockwise direction. If $C$ is a regular closed curve in the plane, i.e., if $C$ is continuously differentiable and has non-vanishing derivative $C ^ { \prime }$, then the degree $D ( C )$ of $C$ is defined to be the winding number of $C ^ { \prime }$. Thus, $D ( C )$ counts the algebraic number of turns of $C ^ { \prime }$ in the counterclockwise direction. | ||
The Whitney–Graustein theorem states that regular homotopy classes of regular curves in the plane are completely classified by the degree, i.e., two regular curves in the plane are regularly homotopic if and only if their degrees are equal (see [[#References|[a2]]]; cf. also [[Homotopy|Homotopy]]). Furthermore, in [[#References|[a2]]] H. Whitney also shows that the degree and the self-intersection number have different parity. | The Whitney–Graustein theorem states that regular homotopy classes of regular curves in the plane are completely classified by the degree, i.e., two regular curves in the plane are regularly homotopic if and only if their degrees are equal (see [[#References|[a2]]]; cf. also [[Homotopy|Homotopy]]). Furthermore, in [[#References|[a2]]] H. Whitney also shows that the degree and the self-intersection number have different parity. | ||
| − | The above definition of degree can be extended to a larger class of curves. If a continuous closed planar curve | + | The above definition of degree can be extended to a larger class of curves. If a continuous closed planar curve $C$ is locally one-to-one, then its secant vectors corresponding to pairs of points on $C$ that are close enough, are non-zero. Therefore, the winding number of the path defined by secant vectors can be used, as before, to define the degree of $C$. More precisely, $D ( C )$ is the winding number of the curve $C ^ { h } : t \rightarrow C ( t + h ) - C ( t ) / h$, for $h$ small enough, or |
| − | + | \begin{equation*} D ( C ) = \operatorname { lim } _ { h \rightarrow 0 } W ( C ^ { h } ). \end{equation*} | |
Replacing regular homotopy by gentle homotopy, i.e., by a homotopy which is locally one-to-one, one gets the following analogue of the Whitney–Graustein Theorem: The gentle homotopy classes of locally one-to-one curves in the plane are completely classified by the degree, i.e., two locally one-to-one curves in the plane are gently homotopic if and only if their degrees are equal (see [[#References|[a1]]]). In this case the degree and the self-intersection number have different parity as well. | Replacing regular homotopy by gentle homotopy, i.e., by a homotopy which is locally one-to-one, one gets the following analogue of the Whitney–Graustein Theorem: The gentle homotopy classes of locally one-to-one curves in the plane are completely classified by the degree, i.e., two locally one-to-one curves in the plane are gently homotopic if and only if their degrees are equal (see [[#References|[a1]]]). In this case the degree and the self-intersection number have different parity as well. | ||
| − | The Whitney–Graustein theorem implies that the tangent vectors of any regular closed curve in the plane point in each direction at least | + | The Whitney–Graustein theorem implies that the tangent vectors of any regular closed curve in the plane point in each direction at least $| D ( C ) |$ times. It is interesting to note that this is true even for differentiable locally one-to-one curves which are not necessarily continuously differentiable (see [[#References|[a1]]]). |
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> M. Kranjc, "Degrees of closed curves in the plane" ''Rocky Mtn. J. Math.'' , '''23''' (1993) pp. 951–978</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> H. Whitney, "On regular closed curves in the plane" ''Comput. Math.'' , '''4''' (1937) pp. 276–284</td></tr></table> |
Latest revision as of 16:52, 1 July 2020
The winding number $W ( C )$ of a continuous closed curve $C$ in the plane, missing the origin, counts the algebraic number of times $C$ goes around the origin in the counterclockwise direction. If $C$ is a regular closed curve in the plane, i.e., if $C$ is continuously differentiable and has non-vanishing derivative $C ^ { \prime }$, then the degree $D ( C )$ of $C$ is defined to be the winding number of $C ^ { \prime }$. Thus, $D ( C )$ counts the algebraic number of turns of $C ^ { \prime }$ in the counterclockwise direction.
The Whitney–Graustein theorem states that regular homotopy classes of regular curves in the plane are completely classified by the degree, i.e., two regular curves in the plane are regularly homotopic if and only if their degrees are equal (see [a2]; cf. also Homotopy). Furthermore, in [a2] H. Whitney also shows that the degree and the self-intersection number have different parity.
The above definition of degree can be extended to a larger class of curves. If a continuous closed planar curve $C$ is locally one-to-one, then its secant vectors corresponding to pairs of points on $C$ that are close enough, are non-zero. Therefore, the winding number of the path defined by secant vectors can be used, as before, to define the degree of $C$. More precisely, $D ( C )$ is the winding number of the curve $C ^ { h } : t \rightarrow C ( t + h ) - C ( t ) / h$, for $h$ small enough, or
\begin{equation*} D ( C ) = \operatorname { lim } _ { h \rightarrow 0 } W ( C ^ { h } ). \end{equation*}
Replacing regular homotopy by gentle homotopy, i.e., by a homotopy which is locally one-to-one, one gets the following analogue of the Whitney–Graustein Theorem: The gentle homotopy classes of locally one-to-one curves in the plane are completely classified by the degree, i.e., two locally one-to-one curves in the plane are gently homotopic if and only if their degrees are equal (see [a1]). In this case the degree and the self-intersection number have different parity as well.
The Whitney–Graustein theorem implies that the tangent vectors of any regular closed curve in the plane point in each direction at least $| D ( C ) |$ times. It is interesting to note that this is true even for differentiable locally one-to-one curves which are not necessarily continuously differentiable (see [a1]).
References
| [a1] | M. Kranjc, "Degrees of closed curves in the plane" Rocky Mtn. J. Math. , 23 (1993) pp. 951–978 |
| [a2] | H. Whitney, "On regular closed curves in the plane" Comput. Math. , 4 (1937) pp. 276–284 |
Whitney-Graustein theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitney-Graustein_theorem&oldid=50051