Related papers: Old and new examples of k-nets in P^2
The paper addresses the $k$-tangle enumeration problem. We introduce a notion of cascade diagram for $k$-tangle projections. An effective enumeration algorithm for projections is proposed based on cascade representation. Tangles projections…
Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into $R^2$, such that no more than two points project to the same point in $R^2$. These diagrams are drawings of 4-regular plane multigraphs. Knots are…
We give a full classification of continuous flexible discrete axial cone-nets, which are called axial C-hedra. The obtained result can also be used to construct their semi-discrete analogs. Moreover, we identify a novel subclass within the…
We study the geometry of $\mathcal{Q}$-conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such $\mathcal{Q}$-conic…
We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set $F$ of edges, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this…
Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.
We define $k$-dimensional digraphs and initiate a study of their spectral theory. The $k$-dimensional digraphs can be viewed as generating graphs for small categories called $k$-graphs. Guided by geometric insight, we obtain several new…
We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$. And we answer a question posed by Meng Chen, whether…
We study linear systems cut out by cones of fixed degree on a smooth complex curve $C\subset\mathbb{P}^{3}$. We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some…
We provide a classification result on nearly free arrangements of lines in the complex projective plane with nodes and triple points.
We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called $k$-transmitters, model a wireless device with a signal that can pass through at most $k$ walls. We show it is NP-hard to…
The projective shape of a configuration of k points or "landmarks" in RP(d) consists of the information that is invariant under projective transformations and hence is reconstructable from uncalibrated camera views. Mathematically, the…
For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore…
In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen…
New results on pentagonal geometries PENT(k,r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3,r) systems with the maximum number of opposite line pairs as well as those…
We study $k$-uniform states in heterogeneous systems whose local dimensions are mixed. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and $k$-uniform…
We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the…
We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and…
A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are…
In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$ As an application one obtains a description…