Related papers: Interlacement and Activities in Delta-Matroids
The width of a delta-matroid is the difference in size between a maximal and minimal feasible set. We give a Rough Structure Theorem for delta-matroids that admit a twist of width one. We apply this theorem to give an excluded minor…
In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated…
We begin with a review of Tutte's homotopy theory, which concerns the structure of certain graph associated to a matroid (together with some extra data). Concretely, Tutte's path theorem asserts that this graph is connected, and his…
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW…
Given a bounded linear operator $T$ on separable Hilbert space, we develop an approach allowing one to construct a matrix representation for $T$ having certain specified algebraic or asymptotic structure. We obtain matrix representations…
Let $\mathcal{M}$ and $\mathcal{N}$ be two matroids on the same ground set $V$. Let $A_1,\dots,A_{2n-1}$ be sets which are independent in both $\mathcal{M}$ and $\mathcal{N}$, satisfying $|A_i|\geq \textrm{min}(i,n)$ for all $i$. We show…
We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method…
Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Pl\"ucker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint…
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the…
We present representative sets-style statements for linear delta-matroids, which are set systems that generalize matroids, with important connections to matching theory and graph embeddings. Furthermore, our proof uses a new approach of…
We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via $(\Delta-Y)$-transformations. We compare this…
This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the…
In this note we introduce a notion of a morphism between two hyperbolic iterated function systems. We prove that the graph of a morphism is the attractor of an iterated function system, giving a Closed Graph Theorem, and show how it can be…
We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids introduced by Bouchet, which naturally arise in topological graph…
We develop certain combinatorial tools for the study of discriminants of general systems of polynomial equations. Applying these tools in a sequel paper, we completely classify components of such discriminants, generalizing the classical…
We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the…
We define a multivariate polynomial that generalizes several interlace polynomials defined by Arratia, Bollobas and Sorkin on the one hand, and Aigner and van der Holst on the other. We follow the route traced by Sokal, who defined a…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which…
The intersection data of a hyperplane arrangement is described by a geometric lattice, or equivalently a simple matroid. There is a rich interplay between this combinatorial structure and the topology of the arrangement complement. In this…