Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

We define a new generalized class of cluster-type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos\pi/n_0+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_0$. In the second part of the paper, we propose the dual graph description of the corresponding Teichmüller spaces, construct the Poisson algebra of the Teichmüller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations thus providing the complete description of the above Teichmüller spaces.