DOI link for Crossing invariants
Crossing invariants book
This chapter sets out containing non-negative integers and possibly the improper element ∞. It discusses the crossing numbers of these knots because it is possible to construct all diagrams with two or fewer virtual crossings. In general, it may be hard to determine the crossing number of a diagram. For virtual knot diagrams, the equivalence class of the connected sum is dependent on the points selected for the splice. For classical knot diagrams, the connected sum always results in the same knot regardless of the location of the sum. The chapter considers equivalence classes of classical link diagrams with no virtual crossings as determined by the classical Reidemeister moves and planar isoptopy and expalins the classical crossing number. It examines two cases: changing the sign of a set of crossings and changing a set of classical crossings into virtual crossings.