A new formula for rotation number Dami´ an Wesenberg October 3, 2020 Abstract We give a new formula for the rotation number (or Whitney index) of a smooth closed plane curve. This formula is obtained from the winding numbers associated with the regions and the crossing points of the curve. One difference with the classic Whitney formula is that ours does not need a base point. 1 Introduction Informally, the rotation number w(γ ) of a regular closed planar curve γ is just the number of complete turns the tangent vector to the curve makes as one passes once around the curve; and the winding number wind(γ,p) of a closed curve γ with respect to a point p is the number of times the curve winds around the point (see Figure 1). In [4] Whitney showed that the rotation number is invariant under regular homotopy. Moreover, again by Whitney [4], there is a simple formula for the rotation number of normal planar curves in terms of the number of positive crossings and negative crossings with respect to a base point. Figure 1: w(γ ) = 1 and wind(γ,p) = 1. In the sections 2, 3 and 4 of this article we give the basic ideas about rotation number and winding numbers, and in section 5 we give our new formula for calculating the rotation number of a closed curve as a function of the winding numbers (Theorem 5.1). 1 arXiv:2010.01422v1 [math.GT] 3 Oct 2020