Polynomial invariants of cyclically ordered graphs

TL;DR

This paper introduces a formal theory of cyclically ordered graphs (cogs), develops various representations, and constructs invariants by adapting known polynomial invariants from related graph theories.

Contribution

It provides the first comprehensive formal framework for cogs and extends polynomial invariants to this new class of graphs.

Findings

Developed multiple representations of cogs.

Constructed new invariants by adapting existing polynomials.

Bridged concepts between topological graph theory and knot theory.

Abstract

Cyclically ordered graphs, or cogs, sit between abstract graphs and cellularly embedded graphs. They arise naturally in topological graph theory, knot theory, and mathematical biology. We develop a formal theory of cogs and establish a number of invariants of cogs. In particular we detail several ways to present cogs and detail how these descriptions can be used to construct cog invariants by adapting the matching, transition and Yamada polynomials.