Related papers: Counting Tripods on the Torus
A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle…
In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…
A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and…
The second part of the paper is devoted to enumeration of $r$-regular toroidal maps up to all homeomorphisms of the torus (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out…
The performance of graph algorithms is often measured in terms of the number of traversed edges per second (TEPS). However, this performance metric is inadequate for a graph operation such as exact triangle counting. In triangle counting,…
Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in…
A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A…
The collection of all topologies on the set of three points is studied treating the topology as quantum-like observable. It turns out to be possible under the assumption of the asymmetry between the spaces of bra- and ket-vectors. The…
Based on previous results of digital topology, this paper focuses on algorithms of topological invariants of objects in 2D and 3D Digital Spaces. We specifically interest in solving hole counting of 2D objects and genus of closed surface in…
We present a new method to count unrooted maps on the sphere up to orientation-preserving homeomorphisms. The principle, called tree-decomposition, is to deform a map into an arborescent structure whose nodes are occupied by constrained…
Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichm\"uller space with the Teichm\"uller metric. In that problem the goal is to count the number of mapping classes that send a given…
Triangle counting is a fundamental problem in the analysis of large graphs. There is a rich body of work on this problem, in varying streaming and distributed models, yet all these algorithms require reading the whole input graph. In many…
Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting…
We develop geometric techniques for counting BPS states in five-dimensional gauge theories engineered by M theory on a toric Calabi-Yau threefold. The problem is approached by studying framed 3d-5d wall-crossing in presence of a single M5…
Those maps of a closed surface to the three-dimensional torus that are homotopic to embeddings are characterized. Particular attention is paid to the somewhat intricate case when the surface is nonorientable.
We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old…
We propose a simple numerical method which computes an approximate value of the winding number of a mapping from 3D torus~$T^3$ to the unitary group~$U(N)$, when $T^3$ is approximated by discrete lattice points. Our method consists of a…
We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main…