On the Theory of Pfaffian Orientations. II. $T$-joins, $k$-cuts, and Duality of Enumeration
This is a continuation of our paper "A Theory of Pfaffian Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of $T$-joins and $k$-cuts of graphs. As a consequence, we show that the computational problem to find the maximum weight of an edge-cut is polynomially solvable for the instances $(G,w)$ where $G$ is a graph embedded on an arbitrary fixed orientable surface and the weight function $w$ has only a bounded number of different values. We also survey the related results concerning a duality of the Tutte polynomial, and present an application for the weight enumerator of a binary code. In a continuation of this paper which is in preparation we present an application to the Ising problem of three-dimensional crystal structures.