Number of Distinct Fragments in Coset Diagrams for PSL2,Z

Coset diagrams [1, 2] are used to demonstrate the graphical representation of the action of the extended modular group PGL2,Z over PLFq=Fq⋃∞. In these sorts of graphs, a closed path of edges and triangles is known as a circuit, and a fragment is emerged by the connection of two or more circuits. The coset diagram evolves through the joining of these fragments. If one vertex of the circuit is fixed by axÏ 1ax−1Ï 2axÏ 3⋯ax−1Ï k∈PSL2,Z, then this circuit is termed to be a length – k circuit, denoted by Ï 1,Ï 2,Ï 3,⋯,Ï k. In this study, we consider two circuits of length −6 as Ω1=α1,α2,α3,α4,α5,α6 and Ω2=β1,β2,β3,β4,β5,β6 with the vertical axis of symmetry that is α2=α6,α3=α5 and β2=β6,β3=β5. It is supposed that Ω is a fragment formed by joining Ω1 and Ω2 at a certain point. The condition for existence of a fragment is given in [3] in the form of a polynomial in Zz. If we change the pair of vertices and connect them, then the resulting fragment and the fragment Ω may coincide. In this article, we find the total number of distinct fragments by joining all the vertices of Ω1 with the vertices of Ω2 provided the condition β4