Planar diagrams for local invariants of graphs in surfaces

TL;DR

This paper introduces a categorical framework for virtual graphs on surfaces, extending classical graph invariants like the flow and Yamada polynomials to nonplanar embeddings, with implications for quantum topology.

Contribution

It defines a planar diagram category for virtual graphs, extends the flow polynomial to the $S$-polynomial, and formulates the $ ext{sl}(N)$ Penrose polynomial for non-cubic graphs, advancing the study of graph invariants in surface embeddings.

Findings

Extension of the flow polynomial to virtual graphs as the $S$-polynomial.

Formulation of the $ ext{sl}(N)$ Penrose polynomial for non-cubic graphs.

A sufficient condition for non-classicality of virtual spatial graphs.

Abstract

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$-polynomial, and formulate the $\mathfrak{sl}(N)$ Penrose polynomial for non-cubic graphs, giving contraction-deletion relations. The $S$-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$-polynomial at squares of integers.