Related papers: Many Touchings Force Many Crossings

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of…

A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes.…

We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve.…

The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…

A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth…

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing…

A closed plane meander of order $n$ is a closed self-avoiding curve intersecting an infinite line $2n$ times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm,…

We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t -7. This improves a result by M. Avendano. Furthermore, we…

A perfect straight-line matching $M$ on a finite set $P$ of points in the plane is a set of segments such that each point in $P$ is an endpoint of exactly one segment. $M$ is non-crossing if no two segments in $M$ cross each other. Given a…

We introduce one dimensional sets to help describe and constrain the integral curves of an $n$ dimensional dynamical system. These curves provide more information about the system than the zero-dimensional sets (fixed points) do. In fact,…

We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.

A thrackle is a drawing of a graph on a surface such that (i) adjacent edges only intersect at their common vertex; and (ii) nonadjacent edges intersect at exactly one point, at which they cross. Conway conjectured that if a graph with $n$…

The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…

Consider two paths $f,g:[0;1]\to [0;1]^2$ on the unit square such that $f(0)=(0,0)$, $f(1)=(1,1)$, $g(0)=(0,1)$, $g(1)=(1,0)$, $f(0;1)\subseteq (0;1)^2$ and $g(0;1)\subseteq (0;1)^2$. By continuity of $f$ and $g$ there is a point of…

We say that a set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set…

For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves…

A plane drawing of a graph is {\em cylindrical} if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The {\em cylindrical crossing…

A \emph{thrackle} is a graph drawn in the plane so that every pair of its edges meet exactly once, either at a common end vertex or in a proper crossing. Conway's thrackle conjecture states that the number of edges is at most the number of…

For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a…