Whitney's formulas for curves on surfaces

TL;DR

This paper generalizes Whitney's classical formula from plane curves to curves on oriented punctured surfaces, incorporating vector fields to define rotation numbers and base point indices, and extends results to non-based curves.

Contribution

It introduces a generalized Whitney formula for curves on punctured surfaces using vector fields, broadening the classical understanding to more complex surfaces.

Findings

Generalized Whitney's formula for punctured surfaces

Defined rotation number and base point index via vector fields

Extended formulas to non-based curves

Abstract

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney's formula to curves on an oriented punctured surface. To define analogs of the rotation number and the index of a base point of a curve, we fix an arbitrary vector field on the surface. Similar formulas are obtained for non-based curves.