Universal Tutte polynomial

Abstract

The Tutte polynomial is a well-studied invariant of graphs and matroids. We first extend the Tutte polynomial from graphs to hypergraphs, and more generally from matroids to polymatroids, as a two-variable polynomial. Our definition is related to previous works of Cameron and Fink and of Kálmán and Postnikov. We then define the universal Tutte polynomial Tn, which is a polynomial of degree n in 2 + (2 n − 1) variables that specializes to the Tutte polynomials of all polymatroids (hence all matroids) on a ground set with n elements. The universal polynomial Tn admits three kinds of symmetries: translation invariance, Sn-invariance, and duality.

Spanning hypertrees, vertex tours and meanders

“…Note that we allow the use of a non-topological contraction only if the resulting graph is not disconnected. For example for γ = (1, 2) and the monopole (σ, α) = ((1, 2, 3, 4), (1, 2)(3, 4)) the pair of permutations (γσ, γα) = ((1)(2, 3, 4), (1)(2)(3, 4)) is not a map: it has two isolated vertices, one of them is incident to the loop (3,4). On the other hand, for γ = (1, 3) and the monopole (σ, α) = ((1, 2, 3, 4), (1, 3)(2, 4)) the pair of permutations (γσ, γα) = ((1, 2)(3, 4), (1)(2, 4)( 3) is a map: we obtain two vertices connected by the edge (2,4).…”

“…Thanks to this duplication, every point of the map may be uniquely described by the ordered pair (i, j) where i is the label of the vertex and j is the label of he edge containing it. For example, after identifying each point with its pair of labels, cycle number 3 of σ is ((3, 7), (3,3), (3,8), (3,9), (3,10), (3,9 ), (3, 10 )), cycle number 5 of α is ((2, 5), (5,5)) and cycle number 9 of α is ((3, 9), (3,9 )). Definition 7.1.…”

“…The permutation θ is a refinement of the permutation σ: for each cycle of σ, the cycles of θ contained in it form a genus zero permutation. For example the cycles of θ contained in the first cycle of σ form the permutation ((1, 1)), ((1, 2), (1,4), (1,3)). We associate a center to each cycle of θ and to each loop edge of α (these are small black disks in Figure 12).…”

“…By merging the edges connecting the centers to the points on their cycles with the edges of our map, we obtain a graph on the centers. For example, in Figure 12 we connect the center of the cycle ((1, 2), (1,4), (1,3)) to the point (1, 2), we continue this edge using the cycle ((1, 2), (4, 2)) of α to the point (4, 2) and then we continue to the center of the cycle ((4, 2)(4, 6)). The resulting topological graph is shown on the right hand side of Figure 12.…”

“…Based on the results in Section 4 there would be several ways to define a Tutte polynomial. We did not commit to any specific choice, because any such definition would depend on the ordering of the hyperedges, unfortunately, just like in the case of the hypergraph Tutte polynomials studied by Bernardi, Kálmán and Postnikov [3] (see also the earlier, less general constructions in [16] and [17]). An important difference between the two approaches is that ours is based on a deletion-contraction process whereas the the hypergraph Tutte polynomials rely on labeling bases in a (poly)matroid.…”