Related papers: Splicing Matroids
We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems…
Let $\mathcal B=\mathcal B_{k,n,p}$ be a random collection of $k$-subsets of $[n]$ where each possible set is present independently with probability $p$. Let $\cal E_{\mathcal B}$ be the event that $\mathcal B$ defines the set of bases of a…
A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We…
We study the representability problem for torsion-free arithmetic matroids. By using a new operation called "reduction" and a "signed Hermite normal form", we provide and implement an algorithm to compute all the representations, up to…
Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar…
Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb P$, and let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is bounded…
Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the…
A bimatroid is a matroid-like generalization of the collection of regular minors of a matrix. In this article, we use the theory of Lorentzian polynomials to study the logarithmic concavity of natural sequences associated to bimatroids.…
One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the…
We characterize the class of threshold matroids by the structure of their defining bases. We also give an example of a shifted matroid which is not threshold, answering a question of Deza and Onn. We conclude by exploring consequences of…
The main goal of this paper is to prove that for odd free knots - that is free knots with all odd crossings - the problem of sliceness (the existence of a spanning disc) has an explicit answer based on the pairing of the knot diagram…
In general it is a difficult problem to construct the lattice of submodules $L(M)$ of a given module $M$. In \cite{St} R. P. Stanley outlined a method for constucting a distributive lattice from a knowledge of its join irreducibles. However…
In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of {\em triangles}, for a matroid to have either $U_{2,4}$ or $F_7$ or $M(K_5)$ as a minor.
We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of…
Cyclic flats form a common structural invariant of both matroids and $q$-matroids, determining these objects through their weighted lattices of cyclic flats. In this paper we exploit this perspective to establish a correspondence between…
We introduce the notion of slice depth of a 2-knot K, which is the minimal integer n such that K is n-slice. We give an upper bound for the slice depth of the n-twist spin of a classical knot which belongs to several specific classes,…
We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in…
The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of ``sparse'' paving matroids that are balanced, but at the same time rich…
A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is…
We introduce the families of solvable and nilpotent matroids, examining their realization spaces, closures, and associated matroid and circuit varieties. We study their realizability, as well as the irreducible decomposition of their…