Bikei invariants and Gauss diagrams for virtual knotted surfaces (Q2799082)

A marked vertex diagram, or a ch-diagram, is a planar diagram of a 4-regular spatial graph with a marker at each 4-valent vertex, which presents a knotted surface in \(\mathbb{R}^4\). Marked vertex diagrams are extended to those with virtual crossings, which present virtual knotted surfaces. A biquandle is an algebraic structure satisfying certain conditions. For a diagram of an oriented link or an oriented marked vertex diagram, an assignment of elements of a biquandle \(X\) to the semiarcs (the portions of the diagram between the crossing points), satisfying certain conditions, is called an \(X\)-coloring, or, in this paper, an \(X\)-labeling. For a finite biquandle \(X\), the number of \(X\)-labelings is an invariant for the presented link or knotted surface, called the biquandle counting invariant. For an unoriented link, we can use a special case of a biquandle called a bikei, also known as an involutory biquandle, which gives the bikei counting invariant.NEWLINENEWLINEIn this paper, the authors focus on bikeis, and extend the bikei counting invariants to those for unorientable marked vertex diagrams. A virtual knot diagram is presented by a Gauss diagram. They introduce the notion of a Gauss diagram for a marked vertex diagram, using a remark that a marked vertex diagram can be transformed to have a single naïve component, and they give a characterization of its orientability.