Graphical virtual links and a polynomial for signed cyclic graphs

Abstract

For a signed cyclic graph [Formula: see text], we can construct a unique virtual link [Formula: see text] by taking the medial construction and converting 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 this paper, we shall prove that a virtual link [Formula: see text] is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial [Formula: see text] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between [Formula: see text] of a signed cyclic graph [Formula: see text] and the bracket polynomial of one of the virtual link diagrams associated with [Formula: see text]. Finally, we give a spanning subgraph expansion for [Formula: see text].