On the Treewidth of Triangulated 3-Manifolds
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Unknot Recognition Through Quantifier Elimination
UNKNOT RECOGNITION THROUGH QUANTIFIER ELIMINATION SYED M. MEESUM AND T. V. H. PRATHAMESH Abstract. Unknot recognition is one of the fundamental questions in low dimensional topology. In this work, we show that this problem can be encoded as a validity problem in the existential fragment of the first-order theory of real closed fields. This encoding is derived using a well-known result on SU(2) representations of knot groups by Kronheimer-Mrowka. We further show that applying existential quantifier elimination to the encoding enables an UnKnot Recogntion algorithm with a complexity of the order 2O(n), where n is the number of crossings in the given knot diagram. Our algorithm is simple to describe and has the same runtime as the currently best known unknot recognition algorithms. 1. Introduction In mathematics, a knot refers to an entangled loop. The fundamental problem in the study of knots is the question of knot recognition: can two given knots be transformed to each other without involving any cutting and pasting? This problem was shown to be decidable by Haken in 1962 [6] using the theory of normal surfaces. We study the special case in which we ask if it is possible to untangle a given knot to an unknot. The UnKnot Recogntion recognition algorithm takes a knot presentation as an input and answers Yes if and only if the given knot can be untangled to an unknot. The best known complexity of UnKnot Recogntion recognition is 2O(n), where n is the number of crossings in a knot diagram [2, 6]. More recent developments show that the UnKnot Recogntion recogni- tion is in NP \ co-NP. -
Computational Topology and the Unique Games Conjecture
Computational Topology and the Unique Games Conjecture Joshua A. Grochow1 Department of Computer Science & Department of Mathematics University of Colorado, Boulder, CO, USA [email protected] https://orcid.org/0000-0002-6466-0476 Jamie Tucker-Foltz2 Amherst College, Amherst, MA, USA [email protected] Abstract Covering spaces of graphs have long been useful for studying expanders (as “graph lifts”) and unique games (as the “label-extended graph”). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial’s 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture – Unique Games and Max-2Lin – are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot–Kindler–Mossel–O’Donnell, FOCS ’04; SICOMP ’07) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new “Unique Games-complete” problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproxim- ability over Z2, and we show Unique Games-completeness over Zk for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian – or more generally for principal G-bundles – Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization. -
Algebraic Topology
Algebraic Topology Vanessa Robins Department of Applied Mathematics Research School of Physics and Engineering The Australian National University Canberra ACT 0200, Australia. email: [email protected] September 11, 2013 Abstract This manuscript will be published as Chapter 5 in Wiley's textbook Mathe- matical Tools for Physicists, 2nd edition, edited by Michael Grinfeld from the University of Strathclyde. The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology. arXiv:1304.7846v2 [math-ph] 10 Sep 2013 1 Contents 1 Introduction 3 2 Homotopy Theory 4 2.1 Homotopy of paths . 4 2.2 The fundamental group . 5 2.3 Homotopy of spaces . 7 2.4 Examples . 7 2.5 Covering spaces . 9 2.6 Extensions and applications . 9 3 Homology 11 3.1 Simplicial complexes . 12 3.2 Simplicial homology groups . 12 3.3 Basic properties of homology groups . 14 3.4 Homological algebra . 16 3.5 Other homology theories . 18 4 Cohomology 18 4.1 De Rham cohomology . 20 5 Morse theory 21 5.1 Basic results . 21 5.2 Extensions and applications . 23 5.3 Forman's discrete Morse theory . 24 6 Computational topology 25 6.1 The fundamental group of a simplicial complex . 26 6.2 Smith normal form for homology . 27 6.3 Persistent homology . 28 6.4 Cell complexes from data . 29 2 1 Introduction Topology is the study of those aspects of shape and structure that do not de- pend on precise knowledge of an object's geometry. -
Ma/CS 6B Class 8: Planar and Plane Graphs
1/26/2017 Ma/CS 6b Class 8: Planar and Plane Graphs By Adam Sheffer The Utilities Problem Problem. There are three houses on a plane. Each needs to be connected to the gas, water, and electricity companies. ◦ Is there a way to make all nine connections without any of the lines crossing each other? ◦ Using a third dimension or sending connections through a company or house is not allowed. 1 1/26/2017 Rephrasing as a Graph How can we rephrase the utilities problem as a graph problem? ◦ Can we draw 퐾3,3 without intersecting edges. Closed Curves A simple closed curve (or a Jordan curve) is a curve that does not cross itself and separates the plane into two regions: the “inside” and the “outside”. 2 1/26/2017 Drawing 퐾3,3 with no Crossings We try to draw 퐾3,3 with no crossings ◦ 퐾3,3 contains a cycle of length six, and it must be drawn as a simple closed curve 퐶. ◦ Each of the remaining three edges is either fully on the inside or fully on the outside of 퐶. 퐾3,3 퐶 No 퐾3,3 Drawing Exists We can only add one red-blue edge inside of 퐶 without crossings. Similarly, we can only add one red-blue edge outside of 퐶 without crossings. Since we need to add three edges, it is impossible to draw 퐾3,3 with no crossings. 퐶 3 1/26/2017 Drawing 퐾4 with no Crossings Can we draw 퐾4 with no crossings? ◦ Yes! Drawing 퐾5 with no Crossings Can we draw 퐾5 with no crossings? ◦ 퐾5 contains a cycle of length five, and it must be drawn as a simple closed curve 퐶. -
Topics in Low Dimensional Computational Topology
THÈSE DE DOCTORAT présentée et soutenue publiquement le 7 juillet 2014 en vue de l’obtention du grade de Docteur de l’École normale supérieure Spécialité : Informatique par ARNAUD DE MESMAY Topics in Low-Dimensional Computational Topology Membres du jury : M. Frédéric CHAZAL (INRIA Saclay – Île de France ) rapporteur M. Éric COLIN DE VERDIÈRE (ENS Paris et CNRS) directeur de thèse M. Jeff ERICKSON (University of Illinois at Urbana-Champaign) rapporteur M. Cyril GAVOILLE (Université de Bordeaux) examinateur M. Pierre PANSU (Université Paris-Sud) examinateur M. Jorge RAMÍREZ-ALFONSÍN (Université Montpellier 2) examinateur Mme Monique TEILLAUD (INRIA Sophia-Antipolis – Méditerranée) examinatrice Autre rapporteur : M. Eric SEDGWICK (DePaul University) Unité mixte de recherche 8548 : Département d’Informatique de l’École normale supérieure École doctorale 386 : Sciences mathématiques de Paris Centre Numéro identifiant de la thèse : 70791 À Monsieur Lagarde, qui m’a donné l’envie d’apprendre. Résumé La topologie, c’est-à-dire l’étude qualitative des formes et des espaces, constitue un domaine classique des mathématiques depuis plus d’un siècle, mais il n’est apparu que récemment que pour de nombreuses applications, il est important de pouvoir calculer in- formatiquement les propriétés topologiques d’un objet. Ce point de vue est la base de la topologie algorithmique, un domaine très actif à l’interface des mathématiques et de l’in- formatique auquel ce travail se rattache. Les trois contributions de cette thèse concernent le développement et l’étude d’algorithmes topologiques pour calculer des décompositions et des déformations d’objets de basse dimension, comme des graphes, des surfaces ou des 3-variétés. -
Planarity and Duality of Finite and Infinite Graphs
JOURNAL OF COMBINATORIAL THEORY, Series B 29, 244-271 (1980) Planarity and Duality of Finite and Infinite Graphs CARSTEN THOMASSEN Matematisk Institut, Universitets Parken, 8000 Aarhus C, Denmark Communicated by the Editors Received September 17, 1979 We present a short proof of the following theorems simultaneously: Kuratowski’s theorem, Fary’s theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski’s theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whit- ney, MacLane, Tutte, and Fournier (in the case of Whitney’s theorem and MacLane’s theorem this has already been done by Tutte). In connection with Tutte’s planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte’s condition in terms of overlap graphs is equivalent to Kuratowski’s condition, we characterize completely the infinite graphs satisfying MacLane’s condition and we prove that the 3- connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte’s criterion involving non-separating cy- cles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs. -
25 HIGH-DIMENSIONAL TOPOLOGICAL DATA ANALYSIS Fr´Ed´Ericchazal
25 HIGH-DIMENSIONAL TOPOLOGICAL DATA ANALYSIS Fr´ed´ericChazal INTRODUCTION Modern data often come as point clouds embedded in high-dimensional Euclidean spaces, or possibly more general metric spaces. They are usually not distributed uniformly, but lie around some highly nonlinear geometric structures with nontriv- ial topology. Topological data analysis (TDA) is an emerging field whose goal is to provide mathematical and algorithmic tools to understand the topological and geometric structure of data. This chapter provides a short introduction to this new field through a few selected topics. The focus is deliberately put on the mathe- matical foundations rather than specific applications, with a particular attention to stability results asserting the relevance of the topological information inferred from data. The chapter is organized in four sections. Section 25.1 is dedicated to distance- based approaches that establish the link between TDA and curve and surface re- construction in computational geometry. Section 25.2 considers homology inference problems and introduces the idea of interleaving of spaces and filtrations, a funda- mental notion in TDA. Section 25.3 is dedicated to the use of persistent homology and its stability properties to design robust topological estimators in TDA. Sec- tion 25.4 briefly presents a few other settings and applications of TDA, including dimensionality reduction, visualization and simplification of data. 25.1 GEOMETRIC INFERENCE AND RECONSTRUCTION Topologically correct reconstruction of geometric shapes from point clouds is a classical problem in computational geometry. The case of smooth curve and surface reconstruction in R3 has been widely studied over the last two decades and has given rise to a wide range of efficient tools and results that are specific to dimensions 2 and 3; see Chapter 35. -
CMSC 754: Lecture 18 Introduction to Computational Topology
CMSC 754 Ahmed Abdelkader (Guest) CMSC 754: Lecture 18 Introduction to Computational Topology The introduction presented here is mainly based on Edelsbrunner and Harer's Computational Topology, while drawing doses of inspiration from Ghrist's Elementary Applied Topology; it is mostly self-contained, at the expense of brevity and less rigor here and there to fit the short span of one or two lectures. An excellent textbook to consult is Hatcher's Algebraic Topology, which is freely available online: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. 2 What is Topology? We are all familiar with Euclidean spaces, especially the plane R where we 3 draw our figures and maps, and the physical space R where we actually live and move about. Our direct experiences with these spaces immediately suggest a natural metric structure which we can use to make useful measurements such as distances, areas, and volumes. Intuitively, a metric recognizes which pairs of locations are close or far. In more physical terms, a metric relates to the amount of energy it takes to move a particle of mass from one location to another. If we are able to move particles between a pair of locations, we say the locations are connected, and if the locations are close, we say they are neighbors. In every day life, we frequently rely more upon the abstract notions of neighborhoods and connectedness if we are not immediately concerned with exact measurements. For instance, it is usually not a big deal if we miss the elevator and opt to take the stairs, or miss an exit on the highway and take the next one; these pairs of paths are equivalent if we are not too worried about running late to an important appointment. -
5 Graph Theory
last edited March 21, 2016 5 Graph Theory Graph theory – the mathematical study of how collections of points can be con- nected – is used today to study problems in economics, physics, chemistry, soci- ology, linguistics, epidemiology, communication, and countless other fields. As complex networks play fundamental roles in financial markets, national security, the spread of disease, and other national and global issues, there is tremendous work being done in this very beautiful and evolving subject. The first several sections covered number systems, sets, cardinality, rational and irrational numbers, prime and composites, and several other topics. We also started to learn about the role that definitions play in mathematics, and we have begun to see how mathematicians prove statements called theorems – we’ve even proven some ourselves. At this point we turn our attention to a beautiful topic in modern mathematics called graph theory. Although this area was first introduced in the 18th century, it did not mature into a field of its own until the last fifty or sixty years. Over that time, it has blossomed into one of the most exciting, and practical, areas of mathematical research. Many readers will first associate the word ‘graph’ with the graph of a func- tion, such as that drawn in Figure 4. Although the word graph is commonly Figure 4: The graph of a function y = f(x). used in mathematics in this sense, it is also has a second, unrelated, meaning. Before providing a rigorous definition that we will use later, we begin with a very rough description and some examples. -
SELF-DUAL GRAPHS 1. Self-Duality of Graphs 1.1. Forms of Self
SELF-DUAL GRAPHS BRIGITTE SERVATIUS AND HERMAN SERVATIUS Abstract. We consider the three forms of self-duality that can be exhibited by a planar graph G, map self-duality, graph self-duality and matroid self- duality. We show how these concepts are related with each other and with the connectivity of G. We use the geometry of self-dual polyhedra together with the structure of the cycle matroid to construct all self-dual graphs. 1. Self-Duality of Graphs 1.1. Forms of Self-duality. Given a planar graph G = (V, E), any regular em- bedding of the topological realization of G into the sphere partitions the sphere into regions called the faces of the embedding, and we write the embedded graph, called a map, as M = (V, E, F ). G may have loops and parallel edges. Given a map M, we form the dual map, M ∗ by placing a vertex f ∗ in the center of each face f, and for ∗ each edge e of M bounding two faces f1 and f2, we draw a dual edge e connecting ∗ ∗ the vertices f1 and f2 and crossing e once transversely. Each vertex v of M will then correspond to a face v∗ of M ∗ and we write M ∗ = (F ∗,E∗,V ∗). If the graph G has distinguishable embeddings, then G may have more than one dual graph, see Figure 1. In this example a portion of the map (V, E, F ) is flipped over on a Q ¡@A@ ¡BB Q ¡ sA @ ¡ sB QQ ¨¨ HH H ¨¨PP ¨¨ ¨ H H ¨ H@ HH¨ @ PP ¨ B¨¨ HH HH HH ¨¨ ¨¨ H HH H ¨¨PP ¨ @ Hs ¨s @¨c H c @ Hs ¢¢ HHs @¨c P@Pc¨¨ s s c c c c s s c c c c @ A ¡ @ A ¢ @s@A ¡ s c c @s@A¢ s c c ∗ ∗ ∗ ∗ 0 ∗ 0∗ ∗ ∗ (V, E,s F ) −→ (F ,E ,V ) (V, E,s F ) −→ (F ,E ,V ) Figure 1. -
Master's Thesis
Master's thesis BWO: Wiskunde∗ A search for the regular tessellations of closed hyperbolic surfaces Student: Benny John Aalders 1st supervisor: Prof. dr. Gert Vegter 2nd supervisor: Dr. Alef Sterk 31st August 2017 ∗Part of the master track: Educatie en Communicatie in de Wiskunde en Natuur- wetenschappen Abstract In this thesis we study regular tessellations of closed orientable surfaces of genus 2 and higher. We differentiate between a purely topological setting and a metric setting. In the topological setting we will describe an algorithm that finds all possible regular tessellations. We also provide the output of this algorithm for genera 2 up to and including 10. In the metric setting we will prove that all topological regular tessellations can be realized metrically. Our method provides an alternative to that of Edmonds, Ewing and Kulkarni. Contents 1 Introduction 1 2 Preliminaries 2 2.1 Hyperbolic geometry . .2 2.2 Fundamental domains . .3 2.3 Side pairings . .4 2.4 A brief discussion on Poincar´e'sTheorem . .5 2.5 Riemann Surfaces . .6 2.6 Tessellations . .9 3 Tessellations of a closed orientable genus-2+ surface 14 3.1 Tessellations of a closed orientable genus-2+ surface consisting of one tile . 14 3.2 How to represent a closed orientable genus-2+ surface as a poly- gon. 16 3.3 Find all regular tessellations of a closed orientable genus-2+ surface 17 4 Making metric regular tessellations out of topological regular tessellations 19 4.1 Exploring the possibilities . 19 4.2 Going from topologically regular to metrically regular . 21 Appendices A An Octave script that prints what all possible fp; qg tessellations for some closed orientable genus-g surface are into a file 28 B All regular tessellations of closed orientable surfaces of genus 2 to 10 33 References 52 1 Introduction In this thesis we will show how to find all possible regular tessellations of a genus-g surface, where g ≥ 2 (genus-2+ surfaces for short). -
Matroid Duality from Topological Duality in Surfaces of Nonnegative Euler Characteristic
Wright State University CORE Scholar Mathematics and Statistics Faculty Publications Mathematics and Statistics 9-2002 Matroid Duality from Topological Duality in Surfaces of Nonnegative Euler Characteristic Dan Slilaty Wright State University - Main Campus, [email protected] Follow this and additional works at: https://corescholar.libraries.wright.edu/math Part of the Applied Mathematics Commons, Applied Statistics Commons, and the Mathematics Commons Repository Citation Slilaty, D. (2002). Matroid Duality from Topological Duality in Surfaces of Nonnegative Euler Characteristic. Combinatorics Probability & Computing, 11 (5), 515-528. https://corescholar.libraries.wright.edu/math/23 This Article is brought to you for free and open access by the Mathematics and Statistics department at CORE Scholar. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. Combinatorics, Probability and Computing (2002) 11, 515–528. c 2002 Cambridge University Press DOI: 10.1017/S0963548302005278 Printed in the United Kingdom Matroid Duality from Topological Duality in Surfaces of Nonnegative Euler Characteristic DANIEL C. SLILATY Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA (e-mail: [email protected]) Received 13 June 2001; revised 11 February 2002 Let G be a connected graph that is 2-cell embedded in a surface S, and let G∗ be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G∗.