# Kauffman-Jones polynomial of a curve on a surface

## TL;DR

This paper introduces a new polynomial invariant for curves on surfaces, extending knot invariants to surface topology, and uses it to estimate minimal self-intersection numbers.

## Contribution

It defines a Kauffman-Jones type polynomial for curves on surfaces, providing a new tool for studying curve homotopy classes and their self-intersection properties.

## Key findings

- The polynomial is an invariant of homotopy classes of curves.
- Provides bounds on the minimal self-intersection number using the polynomial's span.
- Offers a chord diagrammatic method for computing the polynomial.

## Abstract

We introduce a Kauffman-Jones type polynomial $\mathcal{L}_{\gamma}(A)$ for a curve $\gamma$ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial $\mathcal{L}_{\gamma}(A)$ is a Laurent polynomial in one variable $A$ and is an invariant of the homotopy class of $\gamma$. As an application, we obtain an estimate in terms of the span of $\mathcal{L}_{\gamma}(A)$ for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of $\mathcal{L}_{\gamma}(A)$ and show some computational results on the span of $\mathcal{L}_{\gamma}(A)$.