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 Joining of circuits in $PSL(2,\mathbb{Z})$-space Bull. Korean Math. Soc. 2015 Vol. 52, No. 6, 2047-2069 https://doi.org/10.4134/BKMS.2015.52.6.2047Published online November 30, 2015 Qaiser Mushtaq and Abdul Razaq Quaid-i-Azam University Islamabad, Quaid-i-Azam University Islamabad Abstract : The coset diagrams are composed of fragments, and the fragments are further composed of circuits at a certain common point. A condition for the existence of a certain fragment $\gamma$ of a coset diagram in a coset diagram is a polynomial $f$ in $\mathbb{Z} \lbrack z].$ In this paper, we answer the question: how many polynomials are obtained from the fragments, evolved by joining the circuits $\left( n,n\right)$ and $\left( m,m\right) ,$ where \$n Keywords : modular group, coset diagrams, projective line over finite field MSC numbers : Primary 05C25; Secondary 20G40 Full-Text :