Graphical virtual links and a polynomial for signed cyclic graphs

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].

On arrow polynomials of checkerboard colorable virtual links

“…Sometimes, compared to all virtual links, the properties of classical links are more easily extended to checkerboard colorable virtual links. For example, Deng, Jin and Kauffman [1] extended the relationship between signed plane graphs and classical link diagrams to signed cyclic graphs and checkerboard colorable virtual link diagrams. Im, Lee and Lee [5] extended the signature, nullity and determinant of classical oriented links to checkerboard colorable oriented virtual links by presenting the Goeritz matrix for checkerboard colorable virtual links.…”

“…Note that the arrow polynomials of virtual knots 4.55, 4.56, 4.59, 4.76, 4.77 containing a summand with K 2 , 4.96 containing a summand with K 3 are against Theorem 4.3 (2), and virtual knot 4.96 is also against Theorem 4.3 (1). So they are all not checkerboard colorable.…”

Characterization of regular checkerboard colourable twisted duals of ribbon graphs

“…In [10], Ellis-Monaghan and Moffatt introduced the concept of twisted duality by combining partial dual and partial Petrial together. It has a number of applications in graph theory, knot theory, matroid theory and so on [5,6,9,12,16,19,20,22,23]. The twisted dual is also closely related to the medial graph.…”