Colorings and doubled colorings of virtual doodles

TL;DR

This paper introduces a new algebraic approach called doodle switch for coloring virtual doodles, defines an invariant based on these colorings, and explores doubled colorings to distinguish virtual doodles.

Contribution

It presents the concept of doodle switch and the notion of doubled colorings, providing new tools for analyzing virtual doodles.

Findings

Defined a new invariant for virtual doodles

Introduced doubled colorings as a novel method

Provided algebraic framework for virtual doodle analysis

Abstract

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.