Cohomology and Immersed Curves

TL;DR

This paper presents a novel cohomology-based approach for classifying immersed curves on surfaces using Gauss codes, extending previous results and offering new criteria for two-colorability with applications to virtual link theory.

Contribution

It introduces a cohomology-theoretic method for classifying immersed curves and generalizes Carter's classification to broader surface types.

Findings

Provides a classification criterion for immersed curves using cohomology.

Extends Carter's classification result to non-oriented surfaces.

Includes an application to twisted virtual link theory.

Abstract

We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an application to twisted virtual link theory.