DOCSLIB.ORG

Introduction to Knot Theory 18/02/2018

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Knot Theory 18/02/2018 Introduction to Knot Theory 18/02/2018 1. Knots and knot diagrams 1.1. Embeddings and isotopies. LetX andY be Hausdorff topo- logical spaces. Definition 1.1. A continuous mapf:Y→X is an embedding if it is injective and closed. Recall that every functorf:Y→X is closed ifY is compact. Definition 1.2. An embedding Ψ:Y×I→X×I is an isotopy if there is a continuous mapψ:Y×I→X such that Ψ y, t!" ψ y, t!, t!. #e say that embeddingsf,g:Y→X are isotopic if there e$ists an isotopy Ψ:Y×I→X×I% such thatf y!"Ψ y, &! andg y!"Ψ y, 1!. #e 'ill often 'rite Ψt for the mapy 7→Ψ y, t! 'ith ($edt. Definition 1.3. )mbeddingsf,g:Y→X are ambient isotopic if there is an isotopy Ψ:X×I→X×I% such that * 0 " idX andg"Ψ 1 ◦f. +uch * is called an ambient isotopy. ,ot every isotopy can be e$tended to an ambient isotopy. -n particular% there are embeddings that are isotopic% but not ambient isotopic. 1... Knots and links. Definition 1.4. A /not is an embeddingK:S 1 →R 3. #e consider oriented and non-oriented /nots. 0'o /notsK 1 andK 2 are e1uivalent% 'rittenK 1 ∼K 2% if they are ambient isotopic. An embedding ofn disjoint circlesL:S 1 ⊔...⊔S 1 →R 3 is called a lin/. Example 1.5. un/not right trefoil left trefoil (gure-eight 1 2 t'o unlin/ Hopf lin/ 2orromean rings #hitehead lin/ Remark 1.6. 0he 'ord %%ambient3 in 4e(nition 1.5 is important: all /nots are isotopic as embeddings. -ndeed% one can ta/e a ball that contains almost the 'hole /not% and shin/ it to a point: −→ −→ 0here are pathological /nots% such that the follo'ing one: #e 'ant to avoid such /nots. Definition 1.7. A /not6lin/ is tame if it is e1uivalent to a union of (nitely many intervals such /nots6lin/s are called polyhedral or 7L!. 8ther'ise% a /not6lin/ is 'ild. Example 1.8. 9nots and lin/s from )$ample 1.: are tame. ;or e$- ample: ∼ ∼ -n order to avoid 'ild /nots% one can also restrict to smooth embed- dings and smooth isotopies. -n such case% there is no need for ambient isotopies: it is /no'n that smooth embeddings are smoothly isotopic if and only if they are smoothly ambient isotopic. 1.<. Knot diagrams. LetK be a /not of a lin/. 3 Definition 1.9. A projectionπ:R 3 →H onto a planeH is regular if every point fromπ K! is covered by at most t'o points ofK% there are only (nitely many double points% and at each of them the t'o arcs ofK crosses themselves: regular not regular heorem 1.1". Every tame knot/link admits a regular projection. Sketch of Proof. LetK be a tame /not6lin/. Assume (rst that it is polygonal% i.e.K"I 1 ∪ · · · ∪I r% 'here eachI i is a straight segment. =onsider the follo'ing sets: •V 1 is the set of directions of segmentsI i% •V 2 is the set of directions of vectors 'ith origin at a verte$ of K and pointing to a point ofK% •V 3 is the set of directions of lines that intersectK three times. 'here by a direction 'e understand a line going through the origin of 3 R .V 1 is a (nite union of lines%V 2 is contained in a (nite union of planes% 'hereasV 3 is a union of surfaces. Hence% the complement of V :"V 1 ∪V 2 ∪V 3 is dense% in particular not empty% and every point from it is a direction of a regular projection. -n the general case% letK ′ be a polygonal /not6lin/ e1uivalent toK% and 'hich is contained in anǫ-neighborhood ofK for some smallǫ> &. 0hen regular projection ofK are given by points from the complement ofV that are in distance at leastǫ fromV. � Definition 1.11. A diagram of a /not6lin/K is the imageπ K! ofK under a regular projection% 'here at each double point the part ofK closer toH is bro/en. -t follo's from 0heorem 1.1& that tame /nots and lin/s have diagrams. Example 1.12. 0he pictures from )$ample 1.: are /not6lin/ dia- grams. 4 0he main problem of /not theory is to determine 'hether t'o dia- grams represent the same /not6lin/. ;or instance% all the follo'ing are diagrams of the un/not: 0here are certain transformations% called the Reidemeister moves% that change the diagram but preserves the represented /not6lin/: ←−−−→ ←−−−→ R1 R2 ←−−−→ R3 heorem 1.13 Reidemeister!. Link diagramsD 1 andD 2 represent equivalent links if and only ifD 2 can be obtained fromD 1 through finitely many Reidemeister moves. Sketch of Proof. LetL 1 andL 2 be lin/s 'ith diagramsD 1 andD 2 projected on the same planeH. #e may assume thatL 1 andL 2 are polygonal: every tame lin/L is e1uivalent to a polygonal oneL ′% such that their diagrams are ambient isotopic. #e no' use the fact that polygonal lin/s inR 3 are ambient isotopic if and only if they are related by a se1uence of >-move: #hen the moves are su?ciently small% their projections correspond to Reidemeister moves. ;or instance: 5 � Example 1.14. -t is not al'ays an easy tas/ to chec/ if t'o dia- grams represent e1uivalent lin/s. 8ne of the most famous e$amples is the 7er/o pair: -t 'as conjectured for a long time that these diagrams represent differ- ent /nots% until a se1uence of Reidemeister moves have been found by 9eneth 7er/o in 1@A<..
Load more

Recommended publications
  • Introduction to Vassiliev Knot Invariants First Draft. Comments
    Introduction to Vassiliev Knot Invariants First draft. Comments welcome. July 20, 2010 S. Chmutov S. Duzhin J. Mostovoy The Ohio State University, Mansfield Campus, 1680 Univer- sity Drive, Mansfield, OH 44906, USA E-mail address: [email protected] Steklov Institute of Mathematics, St. Petersburg Division, Fontanka 27, St. Petersburg, 191011, Russia E-mail address: [email protected] Departamento de Matematicas,´ CINVESTAV, Apartado Postal 14-740, C.P. 07000 Mexico,´ D.F. Mexico E-mail address: [email protected] Contents Preface 8 Part 1. Fundamentals Chapter 1. Knots and their relatives 15 1.1. Definitions and examples 15 § 1.2. Isotopy 16 § 1.3. Plane knot diagrams 19 § 1.4. Inverses and mirror images 21 § 1.5. Knot tables 23 § 1.6. Algebra of knots 25 § 1.7. Tangles, string links and braids 25 § 1.8. Variations 30 § Exercises 34 Chapter 2. Knot invariants 39 2.1. Definition and first examples 39 § 2.2. Linking number 40 § 2.3. Conway polynomial 43 § 2.4. Jones polynomial 45 § 2.5. Algebra of knot invariants 47 § 2.6. Quantum invariants 47 § 2.7. Two-variable link polynomials 55 § Exercises 62 3 4 Contents Chapter 3. Finite type invariants 69 3.1. Definition of Vassiliev invariants 69 § 3.2. Algebra of Vassiliev invariants 72 § 3.3. Vassiliev invariants of degrees 0, 1 and 2 76 § 3.4. Chord diagrams 78 § 3.5. Invariants of framed knots 80 § 3.6. Classical knot polynomials as Vassiliev invariants 82 § 3.7. Actuality tables 88 § 3.8. Vassiliev invariants of tangles 91 § Exercises 93 Chapter 4.
    [Show full text]
  • On Framings of Knots in 3-Manifolds
    ON FRAMINGS OF KNOTS IN 3-MANIFOLDS RHEA PALAK BAKSHI, DIONNE IBARRA, GABRIEL MONTOYA-VEGA, JÓZEF H. PRZYTYCKI, AND DEBORAH WEEKS Abstract. We show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented 3-manifold is when the manifold has a properly embedded non-separating S2. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough’s work on the mapping class groups of 3-manifolds. We also relate our results to the theory of skein modules. Contents 1. Introduction1 1.1. History of the problem2 2. Preliminaries3 3. Main Results4 3.1. Proofs of the Main Theorems5 3.2. Spin structures and framings7 4. Ramications and Connections to Skein Modules8 4.1. From the Kauman bracket skein module to spin twisted homology9 4.2. The q-homology skein module9 5. Future Directions 10 6. Acknowledgements 11 References 11 1. Introduction We show that the only way to change the framing of a knot in an oriented 3-manifold by ambient isotopy is when the manifold has a properly embedded non-separating S2. More precisely the only change of framing is by the light bulb trick as illustrated in the Figure1. Here the change arXiv:2001.07782v1 [math.GT] 21 Jan 2020 of framing is very local (takes part in S2 × »0; 1¼ embedded in the manifold) and is related to the fact that the fundamental group of SO¹3º is Z2. Furthermore, we use the fact that 3-manifolds possess spin structures given by the parallelization of their tangent bundles.
    [Show full text]
  • How Can We Say 2 Knots Are Not the Same?
    How can we say 2 knots are not the same? SHRUTHI SRIDHAR What’s a knot? A knot is a smooth embedding of the circle S1 in IR3. A link is a smooth embedding of the disjoint union of more than one circle Intuitively, it’s a string knotted up with ends joined up. We represent it on a plane using curves and ‘crossings’. The unknot A ‘figure-8’ knot A ‘wild’ knot (not a knot for us) Hopf Link Two knots or links are the same if they have an ambient isotopy between them. Representing a knot Knots are represented on the plane with strands and crossings where 2 strands cross. We call this picture a knot diagram. Knots can have more than one representation. Reidemeister moves Operations on knot diagrams that don’t change the knot or link Reidemeister moves Theorem: (Reidemeister 1926) Two knot diagrams are of the same knot if and only if one can be obtained from the other through a series of Reidemeister moves. Crossing Number The minimum number of crossings required to represent a knot or link is called its crossing number. Knots arranged by crossing number: Knot Invariants A knot/link invariant is a property of a knot/link that is independent of representation. Trivial Examples: • Crossing number • Knot Representations / ~ where 2 representations are equivalent via Reidemester moves Tricolorability We say a knot is tricolorable if the strands in any projection can be colored with 3 colors such that every crossing has 1 or 3 colors and or the coloring uses more than one color.
    [Show full text]
  • Actions of Mapping Class Groups Athanase Papadopoulos
    Actions of mapping class groups Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Actions of mapping class groups. L. Ji, A. Papadopoulos and S.-T. Yau. Handbook of Group Actions, Vol. I, 31, Higher Education Press; International Press, p. 189-248., 2014, Advanced Lectures in Mathematics, 978-7-04-041363-2. hal-01027411 HAL Id: hal-01027411 https://hal.archives-ouvertes.fr/hal-01027411 Submitted on 21 Jul 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ACTIONS OF MAPPING CLASS GROUPS ATHANASE PAPADOPOULOS Abstract. This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thurston’s sphere of projective foliations equipped with its projective piecewise-linear structure, the space of unmeasured foliations equipped with the quotient topology, the reduced Bers boundary, and the space of geodesic laminations equipped with the Thurston topology. In the third part, we present some perspectives and open problems on other actions of mapping class groups.
    [Show full text]
  • Textile Research Journal
    Textile Research Journal http://trj.sagepub.com A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures S. Grishanov, V. Meshkov and A. Omelchenko Textile Research Journal 2009; 79; 822 DOI: 10.1177/0040517508096221 The online version of this article can be found at: http://trj.sagepub.com/cgi/content/abstract/79/9/822 Published by: http://www.sagepublications.com Additional services and information for Textile Research Journal can be found at: Email Alerts: http://trj.sagepub.com/cgi/alerts Subscriptions: http://trj.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.co.uk/journalsPermissions.nav Citations http://trj.sagepub.com/cgi/content/refs/79/9/822 Downloaded from http://trj.sagepub.com by Andrew Ranicki on October 11, 2009 Textile Research Journal Article A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures S. Grishanov1 Abstract This paper is the second in the series TEAM Research Group, De Montfort University, Leicester, on topological classification of textile structures. UK The classification problem can be resolved with the aid of invariants used in knot theory for classi- V. Meshkov and A. Omelchenko fication of knots and links. Various numerical and St Petersburg State Polytechnical University, St polynomial invariants are considered in application Petersburg, Russia to textile structures. A new Kauffman-type polyno- mial invariant is constructed for doubly-periodic textile structures. The values of the numerical and polynomial invariants are calculated for some sim- plest doubly-periodic interlaced structures and for some woven and knitted textiles.
    [Show full text]
  • Problems in Low-Dimensional Topology
    Problems in Low-Dimensional Topology Edited by Rob Kirby Berkeley - 22 Dec 95 Contents 1 Knot Theory 7 2 Surfaces 85 3 3-Manifolds 97 4 4-Manifolds 179 5 Miscellany 259 Index of Conjectures 282 Index 284 Old Problem Lists 294 Bibliography 301 1 2 CONTENTS Introduction In April, 1977 when my first problem list [38,Kirby,1978] was finished, a good topologist could reasonably hope to understand the main topics in all of low dimensional topology. But at that time Bill Thurston was already starting to greatly influence the study of 2- and 3-manifolds through the introduction of geometry, especially hyperbolic. Four years later in September, 1981, Mike Freedman turned a subject, topological 4-manifolds, in which we expected no progress for years, into a subject in which it seemed we knew everything. A few months later in spring 1982, Simon Donaldson brought gauge theory to 4-manifolds with the first of a remarkable string of theorems showing that smooth 4-manifolds which might not exist or might not be diffeomorphic, in fact, didn’t and weren’t. Exotic R4’s, the strangest of smooth manifolds, followed. And then in late spring 1984, Vaughan Jones brought us the Jones polynomial and later Witten a host of other topological quantum field theories (TQFT’s). Physics has had for at least two decades a remarkable record for guiding mathematicians to remarkable mathematics (Seiberg–Witten gauge theory, new in October, 1994, is the latest example). Lest one think that progress was only made using non-topological techniques, note that Freedman’s work, and other results like knot complements determining knots (Gordon- Luecke) or the Seifert fibered space conjecture (Mess, Scott, Gabai, Casson & Jungreis) were all or mostly classical topology.
    [Show full text]
  • A Diagrammatic Presentation and Its Characterization of Non-Split Compact Surfaces in the 3-Sphere
    A DIAGRAMMATIC PRESENTATION AND ITS CHARACTERIZATION OF NON-SPLIT COMPACT SURFACES IN THE 3-SPHERE SHOSAKU MATSUZAKI Abstract. We give a presentation for a non-split compact surface embedded in the 3-sphere S3 by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces represented by the diagrams are ambient isotopic in S3. 1. Introduction A fundamental problem in Knot Theory is to classify knots and links up to ambient isotopy in S3. Two knots are equivalent if and only if diagrams of them are related by Reidemeister moves [7]: Reidemeister moves characterize a combinatorial structure of knots. Diagrammatic characterizations for spatial graphs, handlebody-knots and surface-knots are also known, where a spatial graph is a graph in S3, a handlebody-knot is a handlebody in S3, and a surface-knot is a closed surface in S4 (cf. [4], [5], [8]). We often use invariants to distinguish the above knots. Many invariants have been discovered on the basis of diagrammatic characterizations. In this paper, we consider presentation of a compact surface embedded in S3, which we call a spatial surface. For a knot or link, we immediately obtain its diagram by perturbing the z-axis of projection slightly. For a spatial surface, however, perturbing the spatial surface is not enough to present it in a useful form: it may be overlapped and folded complexly by multiple layers in the direction of the z-axis of R3 ⊂ S3.
    [Show full text]
  • Lectures Notes on Knot Theory
    Lectures notes on knot theory Andrew Berger (instructor Chris Gerig) Spring 2016 1 Contents 1 Disclaimer 4 2 1/19/16: Introduction + Motivation 5 3 1/21/16: 2nd class 6 3.1 Logistical things . 6 3.2 Minimal introduction to point-set topology . 6 3.3 Equivalence of knots . 7 3.4 Reidemeister moves . 7 4 1/26/16: recap of the last lecture 10 4.1 Recap of last lecture . 10 4.2 Intro to knot complement . 10 4.3 Hard Unknots . 10 5 1/28/16 12 5.1 Logistical things . 12 5.2 Question from last time . 12 5.3 Connect sum operation, knot cancelling, prime knots . 12 6 2/2/16 14 6.1 Orientations . 14 6.2 Linking number . 14 7 2/4/16 15 7.1 Logistical things . 15 7.2 Seifert Surfaces . 15 7.3 Intro to research . 16 8 2/9/16 { The trefoil is knotted 17 8.1 The trefoil is not the unknot . 17 8.2 Braids . 17 8.2.1 The braid group . 17 9 2/11: Coloring 18 9.1 Logistical happenings . 18 9.2 (Tri)Colorings . 18 10 2/16: π1 19 10.1 Logistical things . 19 10.2 Crash course on the fundamental group . 19 11 2/18: Wirtinger presentation 21 2 12 2/23: POLYNOMIALS 22 12.1 Kauffman bracket polynomial . 22 12.2 Provoked questions . 23 13 2/25 24 13.1 Axioms of the Jones polynomial . 24 13.2 Uniqueness of Jones polynomial . 24 13.3 Just how sensitive is the Jones polynomial .
    [Show full text]
  • An Exploration Into Knot Theory Application Regarding Enzymatic
    AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfllment Of the Requirements for the Degree Master of Science In Mathematics By Albert Joseph Jefferson IV 2019 SIGNATURE PAGE THESIS: AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES AUTHOR: Albert Joseph Jefferson IV DATE SUBMITTED: Fall 2019 Department of Mathematics and Statistics Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. John Rock Mathematics & Statistics Dr. Amber Rosin Mathematics & Statistics ii ACKNOWLEDGMENTS I would frst like to thank my mother, who tirelessly continues to support every goal of mine, regardless of the feld, and whose guidance has been invaluable in my life. I also would like to thank my 3 sisters, each of whom are excellent role models and have given me so much inspiration. A giant thank you to my advisor, Robin Wilson, who has dealt with my procrastination gracefully, as well to my thesis committee for being so fexible with the defense (each of you are forever my mentors and I owe so much to you all). Finally, I’d like to thank my boyfriend who has been my rock through this entire process, I love you so so much. iii ABSTRACT This thesis will explore the applications of knot theory in biology at the graduate level, specifcally knot theory’s application on DNA super coiling, site specifc recombination, and the overall topology of DNA. We will start with a short introduction to knot theory, introduce rational tangles (the foundation for our tangle model for enzymatic reactions on DNA), review a very specifc overview of DNA and site specifc recombination as it pertains to our motivations, and fnally introduce the tangle model for a Tyrosine Recom- binase and its infuence on the topology of DNA following its action on the molecule.
    [Show full text]
  • On the Additivity of Crossing Numbers
    ON THE ADDITIVITY OF CROSSING NUMBERS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Alicia Arrua 2015 SIGNATURE PAGE THESIS: ON THE ADDITIVITY OF CROSSING NUMBERS AUTHOR: Alicia Arrua DATE SUBMITTED: Spring 2015 Mathematics and Statistics Department Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. Greisy Winicki-Landman Mathematics & Statistics Dr. Berit Givens Mathematics & Statistics ii ACKNOWLEDGMENTS This thesis would not have been possible without the invaluable knowledge and guidance from Dr. Robin Wilson. His support throughout this entire experience has been amazing and incredibly helpful. I’d also like to thank Dr. Greisy Winicki- Landman and Dr. Berit Givens for being a part of my thesis committee and offering their support. I’d also like to thank my family for putting up with my late nights of work and motivating me when I needed it. Lastly, thank you to the wonderful friends I’ve made during my time at Cal Poly Pomona, their humor and encouragement aided me more than they know. iii ABSTRACT The additivity of crossing numbers over a composition of links has been an open problem for over one hundred years. It has been proved that the crossing number over alternating links is additive independently in 1987 by Louis Kauffman, Kunio Murasugi, and Morwen Thistlethwaite. Further, Yuanan Diao and Hermann Gru­ ber independently proved that the crossing number is additive over a composition of torus links. In order to investigate the additivity of crossing numbers over a composition of a different class of links, we introduce a tool called the deficiency of a link.
    [Show full text]
  • Linking Numbers in Three-Manifolds
    LINKING NUMBERS IN THREE-MANIFOLDS PATRICIA CAHN AND ALEXANDRA KJUCHUKOVA Abstract. Let M be a connected, closed, oriented three-manifold and K, L two rationally null- homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral 3-fold cover of S3 branched along a knot α ⊂ S3. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α. The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. 1. Introduction The notion of linking number between knots in S3 dates back at least as far as Gauss [13]. More generally, given a closed, oriented three-manifold M and two rationally null-homologous, oriented, simple closed curves K; L ⊂ M, the linking number lk(K; L) is defined as well. It is given by 1 (K · C ), where C is a 2-chain in M with boundary nL, n 2 , and · denotes the signed n L L N intersection number. This linking number is well-defined and symmetric [3]. Let the three-manifold M be presented as a 3-fold irregular dihedral branched cover of S3, branched along a knot. Every closed oriented three-manifold admits such a presentation [16, 17, 21].
    [Show full text]
  • The Isotopy Extension Theorem
    The Isotopy Extension Theorem Once we know that tubular neighborhoods and collar neighborhoods exist, a natural follow – up question is the extent to which they are unique . Even in the simplest cases it is clear that many such neighborhoods exist . For example , if f is a smooth map from RRRn to itself which maps the origin to itself , then one can view f as defining a tubular neighborhood of the origin in RRRn. However , it also seems reasonable to say that each such map is equivalent to the standard tubular neighborhood which is given by the identity map on RRRn. This clarifies the goal , which is to define a suitable equivalence relationship on tubular neighborhoods such that any two are equivalent . Before doing this , we shall digress to discuss a concept which enters into this equivalence relation. Definition. Let M and N be two smooth manifolds, and let f and g be smooth embeddings of M into N. We shall say that f and g are isotopic if there is a smooth map H: M × (– εεε, 1 + εεε) →→→ N such that H(x, 0) = f (x) and H(x, 1) = g(x) for all x. Less formally, the map H, which is called an isotopy , is a smooth homotopy from f to g through smooth embeddings . The notion of isotopy defines an equivalence relation on the set of all smooth embeddings from one manifold M to another one N; the reflexive and symmetric properties follow immediately , and the transitive property is a consequence of the following exercise . Exercise. Suppose that f and g are isotopic as above .
    [Show full text]