Related papers: Danzer's configuration revisited

We construct a mathematical framework for twisted N=2 supersymmetric topological quantum field theory on a 4-manifold. Supersymmetry in flat space is defined and the twist homomorphism is constructed, giving us a supermanifold that is the…

Let $G=(V,E)$ be a finite, simple, connected, combinatorial graph on $n$ vertices and let $D \in \mathbb{R}^{n \times n}$ be its graph distance matrix $D_{ij} = d(v_i, v_j)$. Steinerberger (J. Graph Theory, 2023) empirically observed that…

We show that finding orthogonal grid-embeddings of plane graphs (planar with fixed combinatorial embedding) with the minimum number of bends in the so-called Kandinsky model (which allows vertices of degree $> 4$) is NP-complete, thus…

We study the problem of reconfiguring odd matchings, that is, matchings that cover all but a single vertex. Our reconfiguration operation is a so-called flip where the unmatched vertex of the first matching gets matched, while consequently…

In this paper we complete the classification of topological symmetry groups for complete graphs $K_n$ by characterizing which $K_n$ can have a cyclic group, a dihedral group, or a subgroup of $D_m \times D_m$ where $m$ is odd, as its…

If $R$ is a commutative ring, $M$ a compact $R$-oriented manifold and $G$ a finite graph without loops or multiple edges, we consider the graph configuration space $M^G$ and a Bendersky-Gitler type spectral sequence converging to the…

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or "ADE" Dynkin…

The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry…

Let $D_{n,\gamma}$ be the complex of graphs on $n$ vertices and domination number at least $\gamma$. We prove that $D_{n,n-2}$ has the homotopy type of a finite wedge of 2-spheres. This is done by using discrete Morse theory techniques.…

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…

Let $P$ be a set of $n \geq 5$ points in convex position in the plane. The path graph $G(P)$ of $P$ is an abstract graph whose vertices are non-crossing spanning paths of $P$, such that two paths are adjacent if one can be obtained from the…

G\"unter Ziegler has shown in 1989 that some homological invariants associated with the free resolutions of Jacobian ideals of line arrangements are not determined by combinatorics. His classical example involves hexagons inscribed in…

We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is ageneralization to arbitrary…

We extend a result of M. Katz on conformal systoles to all four-manifolds with b^+=1 which have odd intersection form. The same result holds for all four-manifolds with b^+=1 with even intersection form and which are symplectic or satisfy…

Graphical designs are subsets of vertices of a graph that perfectly average a selected set of eigenvectors of the Graph Laplacian. We show that in highly-structured graphs, graphical designs can coincide with highly structured and…

The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the…

Kendall's Similarity Shape Theory for constellations of N points in the carrier space $\mathbb{R}^d$ as quotiented by the similarity group was developed for use in Probability and Statistics. It was subsequently shown to reside within…

We investigate the space $C(X)$ of images of linearly embedded skeleta of simplices $X$ in $\mathbb R^n$, for two families of codimension 2 complexes, each ranging over $n$. In the first family, $X=K$ is the $(n-2)$-skeleton of the…

Let $n$ points be in crescent configurations in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$…

Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order $n\in\N, n>1$, ($n$-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies $n$-HSUSY and investigate the structure of the former in the…