Embedded Graphs And Their Polynomials

Knot theory and plane graph theory can be regarded as a special embedded graph theory.Between the end of the last century and the beginning of this century,knot theory was extended to virtual knot theory and spatial graph theory,and plane graph theory was extended to ribbon graph theory,respectively.Ribbon graphs are actually equivalent to cellular embedded graphs.Recently,virtual knot theory and spatial graph theory were unified into virtual spatial graph theory.The developments in classical knot theory and virtual knot theory lead to formation and developments in ribbon graph theory.For example,the notion of partial dual is proposed.This article focuses on the above types of embedded graphs and their polynomials.The main work and innovation of this article are as follows:(1)Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph.Then Metsidik and Jin charac-terized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph.We first extend Huggett and Moffatt’s work to any orientable ribbon graph,and provide an example to show that it is not true for non-orientable ribbon graphs.Then we introduce the notion of crossing-total directions of a medial graph.And we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph,which are much more simple than Metsidik and Jin’s semi-crossing directions.(2)Kauffman defined the notion of graphical virtual links.We prove that a virtual link is graphical if and only if it is checkerboard colorable.We introduce marking edge operation,and define a F-polynomial for signed cyclic graphs according to deletion-marking edge operation.The relation between the Tutte polynomial of signed planes and the bracket polynomial of link diagrams is extended to the F-polynomial of signed cyclic graphs and the bracket polynomial of virtual link diagrams.We find that the F-polynomial is a special case of the signed Bollobas-Riordan polynomial.(3)Yamada defined the Yamada polynomial of a spatial graph diagram based on the Negami polynomial.We define the generalized Yamada polynomial of a virtual spatial graph diagram in recursive way.We prove that it can be normalized to be a rigid vertex isotopic invariant of virtual spatial graphs and to be a pliable vertex iso-topic invariant for virtual spatial graphs with maximum degree at most 3,respectively.Next,for classical link diagrams,we discuss the relationship between the generalized Yamada polynomial and the Dubrovnik polynomial.And we use the generalized Ya-mada polynomial to detect non-classicality of virtual links.Finally,via Jones-Wenzl operator P2 acting on a virtual spatial graph diagram D,we prove that the 2-cabled bracket polynomial of D with operator P2 is a specialization of the generalized Yama-da polynomial of D.(4)We first discuss the relationship between the flow polynomial and the Kauffman-Vogel polynomial.Jaeger proposed the even subgraph expansions for the flow polynomial of cubic plane graphs.We extend Jaeger’s work to plane graphs with maximum degree at most 4 by introducing even subgraphs with splitting systems.Fi-nally,for the flow polynomial of 4-regular plane graphs,according to the unoriented even subgraph expansion,we obtain formulas of the flow polynomial at 0 and 4,re-spectively.