Homotopies using conformal transformation with invariant total normal twist

The aim of this paper is to detect a homotopy to a spherical curve with invariant total normal twist 0. Also, we use conformal transformation to prove a theorem that for any immersed closed C^3 -curve c in the Euclidean 3-space E^3 with vanishing total normal twist, there exists a Frenet curve homotopic to c in an arbitrary neighborhood of c such that the total normal twist is invariant along the homotopy between c and in that neighborhood.