Related papers: The 3-symmetric Pseudolinear Crossing Number of $K…
Even the most superficial glance at the vast majority of crossing-minimal geometric drawings of $K_n$ reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply {\em 3-symmetric}) . And second, they…
A drawing of a graph in the plane is {\it pseudolinear} if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines, that is, any pair of edges crosses precisely once. A special case are…
We determine ${\bar{\rm{lcr}}}(K_n)$, the rectilinear local crossing number of the complete graph $K_n$ for every $n$. More precisely, for every $n \notin \{8, 14 \}, $ \[ {\bar{\rm{lcr}}}(K_n)=\left\lceil \frac{1}{2} \left( n-3-\left\lceil…
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove…
The point sets of all known optimal rectilinear drawings of $K_n$ share an unmistakeable clustering property, the so--called {\em 3--decomposability}. It is widely believed that the underlying point sets of all optimal rectilinear drawings…
A drawing of a graph is {\em pseudolinear} if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The {\em pseudolinear crossing number} of a graph $G$ is the minimum number of pairwise…
There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings has exactly $3\binom{k+2}{2}$ $(\le k)$-edges, for all $0\le k <…
Since the crossing number of K_{12} is now known to be 150, it is well-known that simple counting arguments and Kleitman's parity theorem for the crossing number of K_{2n+1} combine with a specific drawing of K_{13} to show that the…
The simultaneous crossing number is a new knot invariant which is defined for strongly invertible knots having diagrams with two orthogonal transvergent axes of strong inversions. Because the composition of the two inversions gives a cyclic…
We improve the lower bound on the $d$-dimensional rectilinear crossing number of the complete $d$-uniform hypergraph having $2d$ vertices to $\Omega\left(\dfrac{(4\sqrt{2}/3^{3/4})^d}{d}\right)$ from $\Omega(2^d \sqrt{d})$. We also…
Beginning with the concepts of orientation for a 3-hypergraph and transitivity for an oriented 3-hypergraph, it is natural to study the class of comparability 3-hypergraphs (those that can be transitively oriented). In this work we show…
In this paper we show that a third order PDE system that is a general form of a CR-geometry PDE system has at most a ten-dimensional Lie symmetry algebra. We also show that this estimate is precise.
Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in…
Let $M$ be a connected, closed, oriented three-manifold and $K$, $L$ two rationally null-homologous oriented simple closed curves in $M$. We give an explicit algorithm for computing the linking number between $K$ and $L$ in terms of a…
We prove that the complement ${\mathcal S}:={\mathbb R}^3\setminus{\mathcal K}$ of a 3-dimensional convex polyhedron ${\mathcal K}\subset{\mathbb R}^3$ and its closure $\overline{{\mathcal S}}$ are polynomial images of ${\mathbb R}^3$. The…
Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show…
We prove the 3-fold DT/PT correspondence for K-theoretic vertices via wall-crossing techniques. We provide two different setups, following Mochizuki and following Joyce; both reduce the problem to q-combinatorial identities on word…
A drawing of a graph G in the plane is said to be a rectilinear drawing of G if the edges are required to be line segments (as opposed to Jordan curves). We assume no three vertices are collinear. The rectilinear crossing number of G is the…
In this paper we enumerate $k$-noncrossing RNA pseudoknot structures with given minimum arc- and stack-length. That is, we study the numbers of RNA pseudoknot structures with arc-length $\ge 3$, stack-length $\ge \sigma$ and in which there…
A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normal $d$-pseudomanifolds form a broader class than…