A categorification for the partial-dual genus polynomial

TL;DR

This paper introduces a categorification of the partial-dual genus polynomial for ribbon graphs using an extended Frobenius algebra linked to unoriented topological quantum field theory, providing a new algebraic perspective.

Contribution

It presents the first categorification of the partial-dual genus polynomial, connecting ribbon graph invariants with algebraic structures from topological quantum field theory.

Findings

Provides a new algebraic framework for the partial-dual genus polynomial

Establishes a link between ribbon graph invariants and topological quantum field theory

Offers potential for new computational and theoretical tools in graph topology

Abstract

The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ of a ribbon graph $G$ is the generating function that enumerates all partial duals of $G$. In this paper, we give a categorification for this polynomial. The key ingredient of the construction is an extended Frobenius algebra related to unoriented topological quantum field theory.