Recipe theorems for polynomial invariants on ribbon graphs with half-edges

TL;DR

This paper develops recipe theorems for polynomial invariants of ribbon graphs with half-edges, introducing a generalized transition polynomial and linking it to the Bollabás-Riordan polynomial.

Contribution

It extends polynomial invariants to ribbon graphs with half-edges and establishes a relationship between the transition polynomial and the Bollabás-Riordan polynomial.

Findings

Recipe theorems for the Bollabás-Riordan polynomial on ribbon graphs with half-edges

Definition of a generalized transition polynomial Q

Relationship established between Q and the polynomial al R

Abstract

We provide recipe theorems for the Bollob\`as and Riordan polynomial $\mathcal{R}$ defined on classes of ribbon graphs with half-edges introduced in arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial $Q$ on this new category of ribbon graphs and establish a relationship between $Q$ and $\mathcal{R}$.