Planar diagrams for local invariants of graphs in surfaces

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 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 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.

Graphoids

“…Arguably, one of the most utilized polynomials to study classical spatial graphs is the Yamada polynomial. There are three polynomials for virtual spatial graphs related to the Yamada polynomial that one can turn into graphoid invariants [8,10,20]. We will focus mainly on Fleming-Mellor's formulation R(G) and Deng-Jin-Kauffman's formulation R(G; A, x).…”

The generalized Yamada polynomials of virtual spatial graphs

“…Very recently, Snyder and Miller [24] defined the notion of a virtual graph (equivalently, cyclic graph) in a topological way. And they define the S-polynomial for virtual graphs which is a little different from (but similar with) our generalized Yamada polynomial for cyclic graphs.…”

“…For related studies of spatial graphs, please refer to [9,10,20,21,23,25]. For related studies of virtual spatial graphs, please refer to [4,5,19,22,24].…”