A new formula for rotation number

TL;DR

This paper introduces a novel formula for calculating the rotation number of smooth closed plane curves, utilizing winding numbers around regions and crossings, eliminating the need for a base point unlike traditional methods.

Contribution

The paper presents a new base-point-free formula for the rotation number based on winding numbers, offering a different approach from the classic Whitney formula.

Findings

Provides a base-point-free rotation number formula

Utilizes winding numbers around regions and crossings

Simplifies calculation of rotation number

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.