The Whitney-Graustein theorem states that two oriented regular closed curves in the plane are regularly homotopic iff
their unit tangent vector maps have the same
contour winding number about the origin
(Whitney 1937). In his thesis, S. Smale generalized this result to regular closed curves on a 
-manifold for 
(Smale 1959, p. 281).
Whitney-Graustein Theorem
See also
Contour Winding Number, Regular HomotopyExplore with Wolfram|Alpha
References
Smale, S. "A Classification of Immersions of the Two-Sphere." Trans. Amer. Math. Soc. 90, 281-290, 1959. https://doi.org/10.1090/S0002-9947-1959-0104227-9.Whitney, H. "On Regular Closed Curves in the Plane." Compos. Math. 4, 276-284, 1937.Referenced on Wolfram|Alpha
Whitney-Graustein TheoremCite this as:
Weisstein, Eric W. "Whitney-Graustein Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Whitney-GrausteinTheorem.html