Embeddability of Arrangements of Pseudocircles into the Sphere

Embeddability of arrangements of pseudocircles and graphs on surfaces

A pseudocircle is a simple closed curve on some surface; an arrangement of pseudocircles is a collection of pseudocircles that pairwise intersect in exactly two points, at which they cross. Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most four are embeddable into the sphere, and asked if an analogous result holds for embeddability into orientable surfaces of higher genus. We answer this question positively: An arrangement of pseudocircles is embeddable into an orientable surface of genus~$g$ if and only i…

Upper Bounds on the Number of Vertices of Weight <=k in Particular Arrangements of Pseudocircles

In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight <=k in certain arrangements of pseudocircles in the plane. In particular, forbidding certain subarrangements we improve the known bound of 6n-12 (cf. Kedem et al., 1986) for vertices of weight 0 in arrangements of n pseudocircles to 4n-6. In complete arrangements (i.e. arrangements with each two pseudocircles intersecting) we identify two subarrangements of three and…

伪圆排列作为阴影的正向链环

Positive links with arrangements of pseudocircles as shadows

本文研究以伪圆排列为阴影的非等价正向定向链环数量，并对三种不可避免的排列给出精确估计。

Arrangements of Pseudocircles: On Circularizability

An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. In this paper we present the results of the first thorough study of circularizability. We show that there are exactly four non-circularizable arrangements of 5 pseudocircles (one of them was known before). In the set o…

Arrangements of Pseudocircles: Triangles and Drawings

A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $p_3$ in digon-free arrangements of $n$ pairwise intersecting pseudocircles is at least $2n-4$. We present examples to disprove this conjecture. With a recursive construction based on an example with $12$ pseudocircles and $16$ triangles we obtain a family with $p_3(\mathcal{A})/n \to 16/11 = 1.…

The unavoidable arrangements of pseudocircles

It is known that cyclic arrangements are the only {\em unavoidable} simple arrangements of pseudolines: for each fixed $m\ge 1$, every sufficiently large simple arrangement of pseudolines has a cyclic subarrangement of size $m$. In the same spirit, we show that there are three unavoidable arrangements of pseudocircles.