Related papers: Characterization of signed Gauss paragraphs
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain…
In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for…
We prove that finding a $k$-edge induced subgraph is fixed-parameter tractable, thereby answering an open problem of Leizhen Cai. Our algorithm is based on several combinatorial observations, Gauss' famous \emph{Eureka} theorem [Andrews,…
We prove a structural characterization of graphs that forbid a fixed graph $H$ as an immersion and can be embedded in a surface of Euler genus $\gamma$. In particular, we prove that a graph $G$ that excludes some connected graph $H$ as an…
In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean…
Let S be a compact surface with boundary and F be the set of the orbits of a traversing flow on S. If the flow is generic, its orbit space is a spine G of S, namely G is a graph embedded in S and S is a regular neighbourhood of G. Moreover…
A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by…
We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry…
We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of…
In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a…
In this paper, we give polynomial-time algorithms that can take a graph G with a given combinatorial embedding on an orientable surface S of genus g and produce a planar drawing of G in R^2, with a bounding face defined by a polygonal…
A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we…
The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number,…
The problem of computing all maximal induced subgraphs of a graph G that have a graph property P, also called the maximal P-subgraphs problem, is considered. This problem is studied for hereditary, connected-hereditary and rooted-hereditary…
In this paper we construct an example of a complete immersed minimal surface in $\mathbb{R}^3$ of genus one with two embedded catenoid-type ends, one Enneper-type end and total Gauss curvature $-16\pi.$ The proof of the existence of this…
We revisit the classical results of the formation of trapped surfaces for the Einstein vacuum equation relying on the geodesic foliation, rather than the double null foliation used in all previous results, starting with the seminal work of…
We investigate two notions of saturation for partial planar embeddings of maximal planar graphs. Let $G = (V, E) $ be a vertex-labeled maximal planar graph on $ n $ vertices, which by definition has $3n - 6$ edges. We say that a labeled…
We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of…
A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real…
Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi:…