We define a regular homotopy invariant of closed curves on a surface, and give a formula for the rotation number of closed curves on toms, which is analogous to the Whitney formula for planar curves. As an application, we show a necessary condition for a Gauss word to be realized on toms. (1991): 55M25, 55N33. Mathematics Subject Classifications

In 1993, Cairns and Elton gave a condition for an abstract Gauss word to be realized as the Gauss word of a planar curve. In this paper, we extend their result to curves on surfaces of higher genus. In particular, we give a solution in the case of torus T2.