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Polynomial invariants of cyclically ordered graphs

Paul Bratch, M. N. Ellingham, Joanna A. Ellis-Monaghan, Iain Moffatt, Wout Moltmaker

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.

Polynomial invariants of cyclically ordered graphs

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.

Paper Structure

This paper contains 10 sections, 18 theorems, 40 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Cogs in the context of rotation systems
  4. Cogs in the context of ribbon graphs
  5. Cogs in the context of gems
  6. Cog invariants
  7. A cog polynomial via the matching polynomial
  8. A cog polynomial via the transition polynomial
  9. Cog invariants via the Yamada polynomial
  10. Further study

Key Result

Proposition 2.2

Two graph embeddings have the same underlying cog if and only if they are partial Petrials. Hence there is a 1-1 correspondence between partial Petrie duality classes of graph embeddings and cogs.

Figures (10)

  • Figure 1: Forming a partial Petrial $\mathbb{G}^{\tau(e)}$ at an edge $e$ of a ribbon graph $\mathbb{G}$ .
  • Figure 2: A cog with its corresponding gec.
  • Figure 3: A computation of $\mathcal{M}(\mathcal{G};x,y)$.
  • Figure 4: Two gecs that represent nonisomorphic cogs.
  • Figure 5: A gec and a pointed-gec. The pointed-gec arises by contracting the e-edges $e_1, e_3, e_6$ of the gec.
  • ...and 5 more figures

Theorems & Definitions (43)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 33 more