Descartes Circle Theorem, Steiner Porism, and Spherical Designs
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Simple Infinite Presentations for the Mapping Class Group of a Compact
SIMPLE INFINITE PRESENTATIONS FOR THE MAPPING CLASS GROUP OF A COMPACT NON-ORIENTABLE SURFACE RYOMA KOBAYASHI Abstract. Omori and the author [6] have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group. 1. Introduction For g ≥ 1 and n ≥ 0, we denote by Ng,n the closure of a surface obtained by removing disjoint n disks from a connected sum of g real projective planes, and call this surface a compact non-orientable surface of genus g with n boundary components. We can regard Ng,n as a surface obtained by attaching g M¨obius bands to g boundary components of a sphere with g + n boundary components, as shown in Figure 1. We call these attached M¨obius bands crosscaps. Figure 1. A model of a non-orientable surface Ng,n. The mapping class group M(Ng,n) of Ng,n is defined as the group consisting of isotopy classes of all diffeomorphisms of Ng,n which fix the boundary point- wise. M(N1,0) and M(N1,1) are trivial (see [2]). Finite presentations for M(N2,0), M(N2,1), M(N3,0) and M(N4,0) ware given by [9], [1], [14] and [16] respectively. Paris-Szepietowski [13] gave a finite presentation of M(Ng,n) with Dehn twists and arXiv:2009.02843v1 [math.GT] 7 Sep 2020 crosscap transpositions for g + n > 3 with n ≤ 1. Stukow [15] gave another finite presentation of M(Ng,n) with Dehn twists and one crosscap slide for g + n > 3 with n ≤ 1, applying Tietze transformations for the presentation of M(Ng,n) given in [13]. -
Recognizing Surfaces
RECOGNIZING SURFACES Ivo Nikolov and Alexandru I. Suciu Mathematics Department College of Arts and Sciences Northeastern University Abstract The subject of this poster is the interplay between the topology and the combinatorics of surfaces. The main problem of Topology is to classify spaces up to continuous deformations, known as homeomorphisms. Under certain conditions, topological invariants that capture qualitative and quantitative properties of spaces lead to the enumeration of homeomorphism types. Surfaces are some of the simplest, yet most interesting topological objects. The poster focuses on the main topological invariants of two-dimensional manifolds—orientability, number of boundary components, genus, and Euler characteristic—and how these invariants solve the classification problem for compact surfaces. The poster introduces a Java applet that was written in Fall, 1998 as a class project for a Topology I course. It implements an algorithm that determines the homeomorphism type of a closed surface from a combinatorial description as a polygon with edges identified in pairs. The input for the applet is a string of integers, encoding the edge identifications. The output of the applet consists of three topological invariants that completely classify the resulting surface. Topology of Surfaces Topology is the abstraction of certain geometrical ideas, such as continuity and closeness. Roughly speaking, topol- ogy is the exploration of manifolds, and of the properties that remain invariant under continuous, invertible transforma- tions, known as homeomorphisms. The basic problem is to classify manifolds according to homeomorphism type. In higher dimensions, this is an impossible task, but, in low di- mensions, it can be done. Surfaces are some of the simplest, yet most interesting topological objects. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
THE DISJOINT ANNULUS PROPERTY 1. History And
THE DISJOINT ANNULUS PROPERTY SAUL SCHLEIMER Abstract. A Heegaard splitting of a closed, orientable three-manifold satis- ¯es the Disjoint Annulus Property if each handlebody contains an essential annulus and these are disjoint. This paper proves that, for a ¯xed three- manifold, all but ¯nitely many splittings have the disjoint annulus property. As a corollary, all but ¯nitely many splittings have distance three or less, as de¯ned by Hempel. 1. History and overview Great e®ort has been spent on the classi¯cation problem for Heegaard splittings of three-manifolds. Haken's lemma [2], that all splittings of a reducible manifold are themselves reducible, could be considered one of the ¯rst results in this direction. Weak reducibility was introduced by Casson and Gordon [1] as a generalization of reducibility. They concluded that a weakly reducible splitting is either itself reducible or the manifold in question contains an incompressible surface. Thomp- son [16] later de¯ned the disjoint curve property as a further generalization of weak reducibility. She deduced that all splittings of a toroidal three-manifold have the disjoint curve property. Hempel [5] generalized these ideas to obtain the distance of a splitting, de¯ned in terms of the curve complex. He then adapted an argument of Kobayashi [9] to produce examples of splittings of arbitrarily large distance. Hartshorn [3], also following the ideas of [9], proved that Hempel's distance is bounded by twice the genus of any incompressible surface embedded in the given manifold. Here we introduce the twin annulus property (TAP) for Heegaard splittings as well as the weaker notion of the disjoint annulus property (DAP). -
Quasi–Geodesic Flows
Quasi{geodesic flows Maciej Czarnecki UniwersytetL´odzki,Katedra Geometrii ul. Banacha 22, 90-238L´od´z,Poland e-mail: [email protected] 1 Contents 1 Hyperbolic manifolds and hyperbolic groups 4 1.1 Hyperbolic space . 4 1.2 M¨obiustransformations . 5 1.3 Isometries of hyperbolic spaces . 5 1.4 Gromov hyperbolicity . 7 1.5 Hyperbolic surfaces and hyperbolic manifolds . 8 2 Foliations and flows 10 2.1 Foliations . 10 2.2 Flows . 10 2.3 Anosov and pseudo{Anosov flows . 11 2.4 Geodesic and quasi{geodesic flows . 12 3 Compactification of decomposed plane 13 3.1 Circular order . 13 3.2 Construction of universal circle . 13 3.3 Decompositions of the plane and ordering ends . 14 3.4 End compactification . 15 4 Quasi{geodesic flow and its end extension 16 4.1 Product covering . 16 4.2 Compactification of the flow space . 16 4.3 Properties of extension and the Calegari conjecture . 17 5 Non{compact case 18 5.1 Spaces of spheres . 18 5.2 Constant curvature flows on H2 . 18 5.3 Remarks on geodesic flows in H3 . 19 2 Introduction These are cosy notes of Erasmus+ lectures at Universidad de Granada, Spain on April 16{18, 2018 during the author's stay at IEMath{GR. After Danny Calegari and Steven Frankel we describe a structure of quasi{ geodesic flows on 3{dimensional hyperbolic manifolds. A flow in an action of the addirtive group R o on given manifold. We concentrate on closed 3{dimensional hyperbolic manifolds i.e. having locally isometric covering by the hyperbolic space H3. -
Physics 7D Makeup Midterm Solutions
Physics 7D Makeup Midterm Solutions Problem 1 A hollow, conducting sphere with an outer radius of 0:253 m and an inner radius of 0:194 m has a uniform surface charge density of 6:96 × 10−6 C/m2. A charge of −0:510 µC is now introduced into the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere? (b) Calculate the strength of the electric field just outside the sphere. (c) What is the electric flux through a spherical surface just inside the inner surface of the sphere? Solution (a) Because there is no electric field inside a conductor the sphere must acquire a charge of +0:510 µC on its inner surface. Because there is no net flow of charge onto or off of the conductor this charge must come from the outer surface, so that the total charge of the outer 2 surface is now Qi − 0:510µC = σ(4πrout) − 0:510µC = 5:09µC. Now we divide again by the surface area to obtain the new charge density. 5:09 × 10−6C σ = = 6:32 × 10−6C/m2 (1) new 4π(0:253m)2 (b) The electric field just outside the sphere can be determined by Gauss' law once we determine the total charge on the sphere. We've already accounted for both the initial surface charge and the newly introduced point charge in our expression above, which tells us that −6 Qenc = 5:09 × 10 C. Gauss' law then tells us that Qenc 5 E = 2 = 7:15 × 10 N/C (2) 4π0rout (c) Finally we want to find the electric flux through a spherical surface just inside the sphere's cavity. -
Multivariable Calculus Math 21A
Multivariable Calculus Math 21a Harvard University Spring 2004 Oliver Knill These are some class notes distributed in a multivariable calculus course tought in Spring 2004. This was a physics flavored section. Some of the pages were developed as complements to the text and lectures in the years 2000-2004. While some of the pages are proofread pretty well over the years, others were written just the night before class. The last lecture was "calculus beyond calculus". Glued with it after that are some notes from "last hours" from previous semesters. Oliver Knill. 5/7/2004 Lecture 1: VECTORS DOT PRODUCT O. Knill, Math21a VECTOR OPERATIONS: The ad- ~u + ~v = ~v + ~u commutativity dition and scalar multiplication of ~u + (~v + w~) = (~u + ~v) + w~ additive associativity HOMEWORK: Section 10.1: 42,60: Section 10.2: 4,16 vectors satisfy "obvious" properties. ~u + ~0 = ~0 + ~u = ~0 null vector There is no need to memorize them. r (s ~v) = (r s) ~v scalar associativity ∗ ∗ ∗ ∗ VECTORS. Two points P1 = (x1; y1), Q = P2 = (x2; y2) in the plane determine a vector ~v = x2 x1; y2 y1 . We write here for multiplication (r + s)~v = ~v(r + s) distributivity in scalar h − − i ∗ It points from P1 to P2 and we can write P1 + ~v = P2. with a scalar but usually, the multi- r(~v + w~) = r~v + rw~ distributivity in vector COORDINATES. Points P in space are in one to one correspondence to vectors pointing from 0 to P . The plication sign is left out. 1 ~v = ~v the one element numbers ~vi in a vector ~v = (v1; v2) are also called components or of the vector. -
Classification of Surfaces
-8- Classification of Surfaces In Chapters 2–5, we gave a supposedly complete list of symmetry types of repeating patterns on the plane and sphere. Chapters 6–7 justified our method of “counting the cost” of a signature, but we have yet to show that the given signatures are the only possible ones and that the four features we described are the correct features for whichtolook. Anyrepeatingpatterncanbefoldedintoanorbifoldonsome surface. So to prove that our list of possible orbifolds is complete, we only have to show that we’ve considered all possible surfaces. In this chapter we see that any surface can be obtained from a collection of spheres by punching holes that introduce boundaries (∗) and then adding handles (◦) or crosscaps (×). Since all possible sur- faces can be described in this way, we can conclude that all possible orbifolds are obtainable by adding corner points to their boundaries and cone points to their interiors. This will include not only the orbifolds for the spherical and Euclidean patterns we have already considered, but also those for patterns in the hyperbolic plane that we shall consider in Chapter 17. Caps, Crosscaps, Handles, and Cross-Handles Surfaces are often described by identifying some edges of simpler ones. We’ll speak of zipping up zippers. Mathematically, a zipper (“zip-pair”) is a pair of directed edges (these we call zips)thatwe intend to identify. We’ll indicate a pair of such edges with matching arrows: (opposite page) This surface, like all others, is built out of just a few different kinds of pieces— boundaries, handles, and crosscaps. -
Similarities and Conformal Transformations
Similarities and conformal transformations bS 11...'g. M. Coxt~'rEl~ ~Toronto, Canadt,.) ~lb Enrico Bompw~ui on his ~eivutifie .h¢b~lee Summary. - In ordznary Euclidean space, every iaometry theft leaves no point invariaut ks either a screw displacement (~nchtding ¢~ translationas a special case) or a glide reflection. Every other kind of similarity! is a spiral similarity: th~ produc~ o/ a rotatiolt abottt aliuc and ~, d~httatiot~ J~hose ceuter Ites or this liue. In real inversive space (i.e., Ettclideau space pht~ (~ single point at, i ufiuityl, every cow,formal transfer marion is either ¢~, simdarit!! or the prodt~ct of aJ, iuvers,ou and an isometry. This last remark remains valid w, hen the .uttmber o/ dimensions ts increased. In fact, every con. formal transformation of iuversive n-space 1u l2) is expressible as the prodder of r reflections (~nd ~ iut'ersious, where r ~ "u t- 1, s ~ 2, r -~- s ~_ "n -.b 2. i. Introduction.- According to KLEL',"S Erlanger Programm of 1872, the criterion thai, distinguishes one geometry from another is the group of transformations under which the propositions remain true. ]n particular, Euclidean geometry is characterized by the group of similarities, and inversive geometry by the group of circle-preserving (or sphere-preserving) transforma- tions. It is thus natural to study the precise nature of the most general similarity and of the most general sphere-preserving transformation. For simplicity, we shall consider three-dimensional spaces, with a brief remark at the end as to what happens in n-dimensional spaces. 2. Isometries.- Let us begin by recalling some familiar definitions and theorems. -
Conformal Villarceau Rotors
UvA-DARE (Digital Academic Repository) Conformal Villarceau Rotors Dorst, L. DOI 10.1007/s00006-019-0960-5 Publication date 2019 Document Version Final published version Published in Advances in Applied Clifford Algebras License CC BY Link to publication Citation for published version (APA): Dorst, L. (2019). Conformal Villarceau Rotors. Advances in Applied Clifford Algebras, 29(3), [44]. https://doi.org/10.1007/s00006-019-0960-5 General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:29 Sep 2021 Adv. Appl. Clifford Algebras (2019) 29:44 c The Author(s) 2019 0188-7009/030001-20 published online April 30, 2019 Advances in https://doi.org/10.1007/s00006-019-0960-5 Applied Clifford Algebras Conformal Villarceau Rotors Leo Dorst∗ Abstract. -
Chapter 6 Implicit Function Theorem
Implicit function theorem 1 Chapter 6 Implicit function theorem Chapter 5 has introduced us to the concept of manifolds of dimension m contained in Rn. In the present chapter we are going to give the exact de¯nition of such manifolds and also discuss the crucial theorem of the beginnings of this subject. The name of this theorem is the title of this chapter. We de¯nitely want to maintain the following point of view. An m-dimensional manifold M ½ Rn is an object which exists and has various geometric and calculus properties which are inherent to the manifold, and which should not depend on the particular mathematical formulation we use in describing the manifold. Since our goal is to do lots of calculus on M, we need to have formulas we can use in order to do this sort of work. In the very discussion of these methods we shall gain a clear and precise understanding of what a manifold actually is. We have already done this sort of work in the preceding chapter in the case of hypermani- folds. There we discussed the intrinsic gradient and the fact that the tangent space at a point of such a manifold has dimension n ¡ 1 etc. We also discussed the version of the implicit function theorem that we needed for the discussion of hypermanifolds. We noticed at that time that we were really always working with only the local description of M, and that we didn't particularly care whether we were able to describe M with formulas that held for the entire manifold. -
Some Definitions: If F Is an Orientable Surface in Orientable 3-Manifold M
Some definitions: If F is an orientable surface in orientable 3-manifold M, then F has a collar neighborhood F × I ⊂ M. F has two sides. Can push F (or portion of F ) in one direction. M is prime if every separating sphere bounds a ball. M is irreducible if every sphere bounds a ball. M irreducible iff M prime or M =∼ S2 × S1. A disjoint union of 2-spheres, S, is independent if no compo- nent of M − S is homeomorphic to a punctured sphere (S3 - disjoint union of balls). F is properly embedded in M if F ∩ ∂M = ∂F . Two surfaces F1 and F2 are parallel in M if they are disjoint and M −(F1 ∪F2) has a component X of the form X = F1 ×I and ∂X = F1 ∪ F2. A compressing disk for surface F in M 3 is a disk D ⊂ M such that D ∩ F = ∂D and ∂D does not bound a disk in F (∂D is essential in F ). Defn: A surface F 2 ⊂ M 3 without S2 or D2 components is incompressible if for each disk D ⊂ M with D ∩ F = ∂D, there exists a disk D0 ⊂ F with ∂D = ∂D0 1 Defn: A ∂ compressing disk for surface F in M 3 is a disk D ⊂ M such that ∂D = α ∪ β, α = D ∩ F , β = D ∩ ∂M and α is essential in F (i.e., α is not parallel to ∂F or equivalently 6 ∃γ ⊂ ∂F such that α ∪ γ = ∂D0 for some disk D0 ⊂ F . Defn: If F has a ∂ compressing disk, then F is ∂ compress- ible.