Matrix formulae and skein relations for cluster algebras from surfaces

This paper concerns cluster algebras with principal coefficients associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [23]. Given any (generalized) arc or loop in the surface - with or without self-intersections - we associate an element of (the fraction field of), using products of elements of PSL₂. We prove an abstract combinatorial result which gives a formula for the number of matchings of a snake or band graph in terms of an appropriate product of 2×2 matrices. We then use this formula to prove that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [22, 23]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [13, 12, 14], who worked in the coefficient-free case. The results of this paper will be used in [23] in order to show that certain collections of arcs and loops comprise a vector-space basis for.