Area-preserving isotopies of self-transverse immersions of S1 in ℝ2

Abstract

Let C and C′ be two smooth self-transverse immersions of S¹ into ℝ². Both C and C′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C′ induces a one-to-one correspondence between the disks of C and C′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C′ is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C′. In this paper we show that this is also a sufficient condition.