Invariants via word for curves and fronts

TL;DR

This paper develops an infinite sequence of invariants for curves on surfaces using word theory, recovering classical invariants and classifying plane closed curves, with extensions to long curves and fronts.

Contribution

It introduces a new method to construct invariants for curves on surfaces using word theory, unifying and extending classical invariants like Arnold's.

Findings

Constructed an infinite sequence of invariants for curves on surfaces.

Reproduced Arnold's basic invariants from the new invariants.

Extended invariants to classify long curves and fronts.

Abstract

We construct the infinite sequence of invariants for curves in surfaces by using word theory that V. Turaev introduced. For plane closed curves, we add some extra terms, e.g. the rotation number. From these modified invariants, we get the Arnold's basic invariants and some other invariants. We also express how these invariants classify plane closed curves. In addition, we consider other classes of plane curves: long curves and fronts.