Tangle number of virtual rational tangles

Abstract

We extend the classical coloring rule to compute the tangle (fraction) number of a virtual rational tangle. We use the contemporary argument used by Kauffman [Virtual knot theory, Eur. J. Comb. 20(7) (1999) 662–690] and show that tangle (fraction) number is a complete invariant. The main purpose of this paper is to provide an alternate yet elementary proof of John Conway’s classification theorem for the virtual rational tangles.