ABSTRACT
This chapter explains an upper bound on the span of the bracket polynomial for diagrams with the boundary property. It introduces an abstraction of the checkerboard coloration for oriented virtual links and use this to calculate the number of components in a state and its dual. The chapter utilizes a variety of techniques, including algorithms, proof by induction, and contradiction. The checkerboard coloration of the oriented diagram in the plane transfers to a checkerboard coloration of an abstract link diagram as shown. A cut point or cut loci is a point on an edge of a virtual link diagram. The cut point subdivides the edge of the virtual link diagram into two edges. In a virtual link diagram with cut points, edges are bounded by classical crossings, a classical crossing and a cut point, or two cut points. The chapter discusses the effect of the diagrammatic moves on the checkerboard framing.
