Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Let G be a matrix and M(G) be the matroid defined by linear dependence on the set E of column vectors of G. Roughly speaking, a parcel is a subset of pairs (f,g) of functions defined on E to a suitable Abelian group A satisfying a coboundary condition (that the difference f−g is a flow over A of G) and a congruence condition (that an algebraic or combinatorial function of f and g, such as the sum of the size of the supports of f and g, satisfies some congruence condition). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals a multiple of an evaluation of the Tutte polynomial of M(G) at a point (u,v), usually with complex coordinates, satisfying (u−1)(v−1)=|A|.