Graphical virtual links and a polynomial of signed cyclic graphs

TL;DR

This paper characterizes graphical virtual links derived from signed cyclic graphs as exactly those that are checkerboard colorable, and introduces a polynomial invariant F[G] relating to virtual link diagrams.

Contribution

It proves the equivalence between graphical virtual links and checkerboard colorability, and introduces a new polynomial F[G] with a spanning subgraph expansion.

Findings

Graphical virtual links are precisely checkerboard colorable.

The polynomial F[G] relates to the bracket polynomial of virtual links.

A spanning subgraph expansion for F[G] is established.

Abstract

For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F[G] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally we give a spanning subgraph expansion for F[G].