The folded ribbon theorem for regular closed curves in the plane

George K. Francis
1968 Bulletin of the American Mathematical Society
Let 5 be the oriented circle with base point, E the oriented Euclidian plane, and V the positively oriented two frames in E. Let L be the space of D-regular immersions g: S-^E with continuous right transverse field |. For gGL, set dg= (g, g, g') : S-+EX V. A positive monotone regular homotopy ( = monotopy) from loop g_i to g+i is a C 1regular homotopy G: [•-1, + l]XS-*E with positive Jacobian and dG(i, x)=dgi(x), i = ±1, where öG=(G, dG/dt, dG/dx). A negative monotopy G from g_i to g+i is such
more » ... g_i to g+i is such that G*(t, x) =G( -/, #) is a positive monotopy from g+i to g_i. A monotopy is stronger than a regular homotopy in that the latter requires only that dG/dx 5^0. The tangent winding number TWN of g in L is the degree of g'/\g'\ : S-+S. Because degree is a homotopy invariant, regular homotopy preserves the TWN. The converse of this is the Whitney-Graustein Theorem  . The TWN actually classifies L in a much stronger fashion. THEOREM. For two regular loops gi, i= ±1, of like TWN, there always is a regular loop g 0 and two monotopies Hi'. gi~go, i= ± 1, of like sign equal to sign (TWN± J). Note that TWN = 0 belongs to both cases. For TWN = 1, two concentric circles are monotopic. Not so for two circles whose interiors are disjoint; yet each is monotopic to a circle surrounding them both. The method of proof is entirely constructive. The normal loops L N have only simple, signed, transverse self-intersections ( = nodes). L N is dense and open ( = generic) in L under the topology induced by ||g -Â|| =max \dg(x)-dh(x)\, xÇzS. (See  for details.) PROPOSITION 1. If gGL and e>0, there is an hGL N with \\g -h\\ <e and a monotopy of prescribed sign between them.