Winding number and isotopies for closed curves on \(S^ 2\)
The author classifies curves on \(S^ 2\) with bounded winding number in the sense of \textit{W. Klingenberg} [Geom. Dedicata 14, 1-11 (1983; Zbl 0509.53002)] by using m, the number of simple constituents of the curve, h the number of connected components of the complement of the curve and k, the number of self-intersections. The winding number w(c) is defined only for curves that either are differentiable immersions or that have only transversal self-intersections. The author proves first that m and w(c) have the same parity, m and k have opposite parity, and some additional technical results. The main result is: If c has only transversal self-intersections and w(c)\(\geq 3\) then c is the union, up to w-preserving isotopies (that may contain curves that violate the transversality condition) to the union of a simple curve and a good curve c' with \(w(c')=w(c)-1\). If \(w(c)=1\) then the curve is isotopic to a great circle. So the only problem left is \(w(c)=2\). In this case, every curve \(c_ i\) of the simple decomposition of c leaves all the others in one of the two connected components of \(S^ 2-c_ i\). Things can get complicated if \(w(c)=2\), \(h\geq 3\); so one has to look for further research to find isotopies that decrease h.
scientific article; zbMATH DE number 3868129
scientific article