Related papers: Linear relations for a generalized Tutte polynomia…
We develop the theory of matroids over one-dimensional algebraic groups, with special emphasis on positive characteristic. In particular, we compute the Lindstr\"om valuations and Frobenius flocks of such matroids. Building on work by Evans…
We consider the problem of approximating certain combinatorial polynomials. First, we consider the problem of approximating the Tutte polynomial of a binary matroid with parameters q>= 2 and gamma. (Relative to the classical (x,y)…
We generalize the well-studied notion of a modular pair of a finite matroid to arbitrary families of sets in infinite matroids, and use it to develop the theory of infinite matroids in several as-yet-unexplored areas. Our results include a…
Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. In the study of the sl2 weight system, it was shown that its value on a…
Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} \Gamma_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $\Gamma_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$…
We give an analogue of the Tutte polynomial for hypermaps. This polynomial can be defined as either a sum over subhypermaps, or recursively through deletion-contraction reductions where the terminal forms consist of isolated vertices. Our…
We generalise the construction of infinite matroids from trees of matroids to allow the matroids at the nodes, as well as the field over which they are represented, to be infinite.
Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of…
We transform Tutte-Grothedieck invariants thus also Tutte polynomials on matroids so that the contraction-deletion rule for loops (isthmuses) coincides with the general case.
Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the…
We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are…
A common generalization for the chromatic polynomial and the flow polynomial of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning for the coefficients of $T$ was discovered by Tutte at the beginning of its…
Let $f(x)$ be a monic polynomial over $\mathbb{Q}$ with complex roots $\alpha_1,\dots,\alpha_n$. Linear relations among them and $1$ over $\mathbb{Q}$ play an important role when we study the distribution of roots modulo a prime. We study…
The Rayleigh property is a negative correlation property of electrical networks, which was generalized to matroids by Choe and Wagner. We prove that positroids, a class of matroid introduced by Postnikov which have seen many recent…
Blasiak verified a conjecture of White for graphic matroids by showing that the toric ideal of a graphic matroid is generated by quadrics. In this paper, we extend this result to frame matroids satisfying a linearity condition. Such classes…
In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and determines thus a finite type…
Graphings serve as limit objects for bounded-degree graphs. We define the ``cycle matroid'' of a graphing as a submodular setfunction, with values in [0,1], which generalizes (up to normalization) the cycle matroid of finite graphs. We…
A topological mating is a map defined by gluing together the filled Julia sets of two quadratic polynomials. The identifications are visualized and understood by pinching ray-equivalence classes of the formal mating. For postcritically…
This article contains general formulas for Tutte and Jones polynomials for families of knots and links given in Conway notation and "portraits of families"-- plots of zeroes of their corresponding Jones polynomials.
Unlike Whitney's definition of the corank-nullity generating function $T(G;x+1,y+1)$, Tutte's definition of his now eponymous polynomial $T(G;x,y)$ requires a total order on the edges of which the polynomial is a posteriori independent.…