Related papers: Generalized Lantern Relations and Planar Line Arra…
We show that in the mapping class group of a surface any relation between Dehn twists of the form T_x^j T_y^k = M (M a multitwist) is the lantern relation, and any relation of the form (T_x T_y)^k = M (where T_x commutes with M) is the…
There are several notions of duality between lines and points. In this note, it is shown that all these can be studied in a unified way. Most interesting properties are independent of specific choices. It is also shown that either dual…
We introduce several new families of relations in the mapping class groups of planar surfaces, each equating two products of right-handed Dehn twists. The interest of these relations lies in their geometric interpretation in terms of…
A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in…
We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]
We describe all inequalities among generalized diagonals in positive semi-definite matrices. These turn out to be governed by a simple partial order on the symmetric group. This provides an analogue of results of Drake, Gerrish, and…
We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances…
We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link…
In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices…
De Bruijn and Erd\H{o}s proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on…
The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the…
We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we give (i) a generalized addition theorem for…
In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially…
A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in…
We study linear relations between face numbers of levels in arrangements. Let $V = \{ v_1, \ldots, v_n \} \subset \mathbf{R}^{r}$ be a vector configuration in general position, and let $\mathcal{A}(V)$ be polar dual arrangement of…
In this paper we present a smallest possible counterexample to the Numerical Terao's Conjecture in the class of line arrangements in the complex projective plane. Our example consists of a pair of two arrangements with $13$ lines. Moreover,…
We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all…
In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of $21$ lines in the complex projective plane.
Comtraces (combined traces) are extensions of Mazurkiewicz traces that can model the "not later than" relationship. In this paper, we first introduce the novel notion of generalized comtraces, extensions of comtraces that can additionally…
This is a short note on various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove…