The Solution of the Ar T-System for Arbitrary Boundary

We present an explicit solution of the Ar T -system for arbitrary boundary conditions. For each boundary, this is done by constructing a network, i.e. a graph with positively weighted edges, and the solution is expressed as the partition function for a family of non-intersecting paths on the network. This proves in particular the positive Laurent property, namely that the solutions are all Laurent polynomials of the initial data with non-negative integer coefficients.