DOCSLIB.ORG

RESEARCH STATEMENT My Approach to Math

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
RESEARCH STATEMENT My Approach to Math RESEARCH STATEMENT THOMAS KINDRED My approach to math research is highly visual, explicitly constructive, often combinatorial, sometimes com- putational: ask intuitive but challenging questions about deep ideas; use elementary methods, especially constructive, visual, and/or combinatorial ones, to attack these questions; persist; engage fully with the writing and revision process, refining insights to a level of clarity where they seem obvious; use this clarity to look for further insights, perhaps computational; share; ask more questions; continue in this way. My goal is not just to prove, but to understand and convey. My research stems, at least indirectly, from a fascination with spanning surfaces in S3, which occupy a perfect intersectional space, relative to my strengths and interests, from which to reach out to diverse areas: • Spanning surfaces are versatile: nearly every aspect of low-dimensional topology can be understood in terms of knots and links in S3, and spanning surfaces offer a visual, constructive, geometric approach to nearly every part of knot theory. The basic idea is that the space around a knotted circle carries a surprising wealth of mathematical data, and spanning surfaces help parse this data. • Spanning surfaces invite explicitly constructive approaches, often involving elementary methods, often combinatorial in nature. These all play to my relative strengths as a researcher. • Spanning surfaces are fun! You draw them. You can hold them in your hands. Their visual appeal conceals their profound mathematical relevance in a way that reminds me, by analogy, of a quotation: Music is the pleasure the human mind experiences from counting without being aware that it is counting { Gottfried Liebniz Organization: Section 1 surveys the mathematical and historical foundations of my research. I hope that mathematicians from all backgrounds find it accessible. It can be treated as a self-contained survey of the main themes of my research. Section 2 surveys my eight completed papers, and my ninth paper, in preparation, and describes ways that I plan to extend them. Section 3 describes other likely directions for my future research. First, though, I want to highlight a recent result which conveys the overall flavor of my research. As I will describe further in xx1-2, in 1993 Menasco and Thistlethwaite used the Jones polynomial to give the first proof of Tait's 1898 flyping conjecture. I recently gave the first purely geometric proof of their theorem: Flyping Theorem: Any two reduced alternating diagrams of a given prime nonsplit link are related by a sequence of flype moves. My proof stems from the insight that a flype move on a diagram corresponds to a geometric operation, which I call re-plumbing, on one of that diagrams' chessboard surfaces (and an isotopy of the other chessboard surface). Figure 1 illustrates this correspondence. This particular insight and the approach to the flyping theorem that grew out of it (and less directly all of my research) all grew out of my first week conducting undergraduate research, with Colin Adams at Williams' REU SMALL in 2005. 1 2 THOMAS KINDRED T T 2 2 2 T T T 1 1 Figure 1. A flype move (top) on a link diagram corresponds to a re-plumbing move (bot- tom) on one of its chessboard surfaces (here, the black surface) and an isotopy of the other surface (here, the white surface). 1. Mathematical and historical background A mathematical knot is an embedded circle in 3-dimensional space S3; a knot consisting of more than one circle is called a link. Most of what follows works for links, but I will focus on knots, where the definitions and theorems tend to be more straightforward. The first knot theorist, Gauss, described how to rep- resent knots diagrammatically, as in Figure 1. Gauss also described how to represent any knot diagram by a sequence of positive integers: pick a basepoint and orient the knot; then walk along the knot, recording each new crossing with the next unused integer and each repeated crossing with the corresponding inte- 1 ger. To this day, the interplay between the combina- Figure 2. An alternating knot dia- torics of knot diagrams and the topology of knots and gram. Using the indicated basepoint manifolds often reveals new deep and surprising con- and orientation, the Gauss code is 2 nections (e.g. see x2.7), and Gauss codes are perfect (1; 2; 3; 1; 4; 5; 6; 3; 2; 4; 7; 6; 5; 7). for computing (see x2.5 for examples). The first systematic treatment of knot theory came almost a century later, when in 1898 Tait constructed a table of all alternating knots through ten crossings. Tait justified his tabulation entirely on intuition (the hard part was proving that knots that seem to be different actually are different). Tait conjectured [64]: • Any reduced3 alternating diagram realizes its knot's (minimal) crossing number. • All reduced alternating diagrams of a given alternating knot have the same writhe.4 1In general, one uses the over/under information at the crossings to attache signs to these integers; when the diagram alternates between over- and under-crossings, this is unnecessary, subject to the convention that the first crossing is an over-crossing. 2and DT codes, which are half the length but carry the same information 3A knot diagram is reduced if it has no \nugatory" crossing: or more generally . 4Orient a knot diagram D arbitrarily. Then every crossing looks like or . The writhe of D is j j − j j. RESEARCH STATEMENT 3 • All reduced alternating diagrams of any prime5 alternating knot are related by flypes (see Fig. 1). Interestingly, Tait was motivated by an erroneous belief in the vortex theory of atoms, which held that atoms were knotted circles in the ether; Tait hoped that his table might correspond to a table of the elements. Before long, physics abandoned the theory of the ether, in favor of Einstein's general theory of relativity, and the vortex theory of atoms, in favor of quantum mechanics. Still, knot theory soon found new motivation in Dehn's insight (circa 1910) that, by drilling out (thickened) knotted circles and gluing them back in \another way," one could construct many different 3-manifolds. In 1960-62, Lickorish and Wallace used Heegaard splittings and mapping class groups of surfaces to prove that every closed orientable 3-manifold can be obtained via such Dehn surgery, thus implying that the study of 3-manifolds corresponds to the study of integer-weighted links in S3; the integer tells one how to glue the thickened knot back in. The advent of Kirby calculus in 1978 created a similar correspondence between closed oriented smooth 4-manifolds and integer-weighted links, this time in connected sums of S1 ×S2. The connection here is that performing Dehn surgery on S3 = @D4 corresponds to attaching a 2-handle to D4. (More on 4-manifolds at the end of x1.) Despite these fantastic reasons to study knots, Tait's conjectures all remained unproven into the mid-80's. This was especially striking, given that his tabulation was verified using discoveries from the 1920's and 30's: • Reidemeister described three types of moves that relate all diagrams of a given knot. These moves provide a rigorous framework for developing knot invariants: numbers, polynomials, groups, etc. that one assigns to a knot in a way that doesn't depend on the diagram. • Seifert proved algorithmically that every knot K is the boundary of a connected orientable (i.e. 2-sided) surface F embedded in S3. Such F is now called a Seifert surface for K. The minimum genus among all Seifert surfaces for K is called its classical genus, denoted g(K). • Alexander and Briggs discovered the first polynomial invariant for knots. One can compute the Alexander polynomial ∆(K) directly from a knot diagram, but ∆(K) has a geometric interpreta- tion: construct the infinite cyclic cover X of S3 nL by cutting along a Seifert surface, taking infinitely −1 many copies, and gluing them together; then H1(X) is isomorphic to Z[t; t ]=∆(K). In addition to verifying Tait's tabulation, these discoveries opened up, and in some cases solved, new ques- tions. For example, in the 1950's, Crowell and Murasugi independently used ∆(K) to prove that applying Seifert's algorithm to any reduced alternating knot diagram yields a surface that realizes g(K) [18, 55]. Meanwhile, the two theories that had displaced the theory of the ether and atomic vortices had come into conflict, creating what to this day remains the biggest open problem in physics: how to \quantize gravity"? This would unify quantum mechanics and relativity, the theories which respectively govern the cosmically large and the immeasurably small. In the context of this conundrum and Tait's still-unproven conjectures, a pair of revolutionary discoveries further ignited interest in the theory of knots. First, in the late 1970's, Thurston realized that \most" knot complements admit a (unique) hyperbolic structure. Hyperbolic geometry, arises naturally in relativity as a \slice of the future," the set of all points with a given time-like separation from, say, here and now. Second, Jones' work in operator algebras (a vital tool in quantum mechanics) led in 1985 to the discovery of a new polynomial invariant for knots [39].6 The Jones polynomial quickly led to several proofs of Tait's conjectures about crossing number and writhe [65, 42, 57, 68]. The proofs were surprisingly straightforward, 5A knot K is prime if it cannot be written as a connected sum, i.e. if, whenever an embedded sphere Q ⊂ S3 intersects K transversally in two points, one of the two arcs of K n Q is unknotted.
Load more

Recommended publications
  • Of the American Mathematical Society August 2017 Volume 64, Number 7
    ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves
    [Show full text]
  • Tait's Flyping Conjecture for 4-Regular Graphs
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series B 95 (2005) 318–332 www.elsevier.com/locate/jctb Tait’s flyping conjecture for 4-regular graphs Jörg Sawollek Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany Received 8 July 1998 Available online 19 July 2005 Abstract Tait’s flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are identical on the class of diagrams considered. © 2005 Elsevier Inc. All rights reserved. Keywords: Knotted graph; Alternating diagram; Flyping conjecture 0. Introduction Very early in the history of knot theory attention has been paid to alternating diagrams of knots and links. At the end of the 19th century Tait [21] stated several famous conjectures on alternating link diagrams that could not be verified for about a century. The conjectures concerning minimal crossing numbers of reduced, alternating link diagrams [15, Theorems A, B] have been proved independently by Thistlethwaite [22], Murasugi [15], and Kauffman [6]. Tait’s flyping conjecture, claiming that two reduced, alternating, prime diagrams of a given link can be connected by a finite sequence of so-called flypes (see [4, p. 311] for Tait’s original terminology), has been shown by Menasco and Thistlethwaite [14], and for a special case, namely, for well-connected diagrams, also by Schrijver [20].
    [Show full text]
  • Non-Orientable Lagrangian Cobordisms Between Legendrian Knots
    NON-ORIENTABLE LAGRANGIAN COBORDISMS BETWEEN LEGENDRIAN KNOTS ORSOLA CAPOVILLA-SEARLE AND LISA TRAYNOR 3 Abstract. In the symplectization of standard contact 3-space, R × R , it is known that an orientable Lagrangian cobordism between a Leg- endrian knot and itself, also known as an orientable Lagrangian endo- cobordism for the Legendrian knot, must have genus 0. We show that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a non-orientable Lagrangian en- docobordism must be a positive multiple of 4. The more restrictive exact, non-orientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do exist for any stabilized Legen- drian knot. Moreover, the relation defined by exact, non-orientable La- grangian cobordism on the set of stabilized Legendrian knots is symmet- ric and defines an equivalence relation, a contrast to the non-symmetric relation defined by orientable Lagrangian cobordisms. 1. Introduction Smooth cobordisms are a common object of study in topology. Motivated by ideas in symplectic field theory, [19], Lagrangian cobordisms that are cylindrical over Legendrian submanifolds outside a compact set have been an active area of research interest. Throughout this paper, we will study 3 Lagrangian cobordisms in the symplectization of the standard contact R , 3 t namely the symplectic manifold (R×R ; d(e α)) where α = dz−ydx, that co- incide with the cylinders R×Λ+ (respectively, R×Λ−) when the R-coordinate is sufficiently positive (respectively, negative). Our focus will be on non- orientable Lagrangian cobordisms between Legendrian knots Λ+ and Λ− and non-orientable Lagrangian endocobordisms, which are non-orientable Lagrangian cobordisms with Λ+ = Λ−.
    [Show full text]
  • Alternating Knots
    ALTERNATING KNOTS WILLIAM W. MENASCO Abstract. This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory. Introduction Figure 1. P.G. Tait's first knot table where he lists all knot types up to 7 crossings. (From reference [6], courtesy of J. Hoste, M. Thistlethwaite and J. Weeks.) 3 ∼ A knot K ⊂ S is alternating if it has a regular planar diagram DK ⊂ P(= S2) ⊂ S3 such that, when traveling around K , the crossings alternate, over-under- over-under, all the way along K in DK . Figure1 show the first 15 knot types in P. G. Tait's earliest table and each diagram exhibits this alternating pattern. This simple arXiv:1901.00582v1 [math.GT] 3 Jan 2019 definition is very unsatisfying. A knot is alternating if we can draw it as an alternating diagram? There is no mention of any geometric structure. Dissatisfied with this characterization of an alternating knot, Ralph Fox (1913-1973) asked: "What is an alternating knot?" black white white black Figure 2. Going from a black to white region near a crossing. 1 2 WILLIAM W. MENASCO Let's make an initial attempt to address this dissatisfaction by giving a different characterization of an alternating diagram that is immediate from the over-under- over-under characterization. As with all regular planar diagrams of knots in S3, the regions of an alternating diagram can be colored in a checkerboard fashion. Thus, at each crossing (see figure2) we will have \two" white regions and \two" black regions coming together with similarly colored regions being kitty-corner to each other.
    [Show full text]
  • Crossing Number of Alternating Knots in S × I
    Pacific Journal of Mathematics CROSSING NUMBER OF ALTERNATING KNOTS IN S × I Colin Adams, Thomas Fleming, Michael Levin, and Ari M. Turner Volume 203 No. 1 March 2002 PACIFIC JOURNAL OF MATHEMATICS Vol. 203, No. 1, 2002 CROSSING NUMBER OF ALTERNATING KNOTS IN S × I Colin Adams, Thomas Fleming, Michael Levin, and Ari M. Turner One of the Tait conjectures, which was stated 100 years ago and proved in the 1980’s, said that reduced alternating projections of alternating knots have the minimal number of crossings. We prove a generalization of this for knots in S ×I, where S is a surface. We use a combination of geometric and polynomial techniques. 1. Introduction. A hundred years ago, Tait conjectured that the number of crossings in a reduced alternating projection of an alternating knot is minimal. This state- ment was proven in 1986 by Kauffman, Murasugi and Thistlethwaite, [6], [10], [11], working independently. Their proofs relied on the new polynomi- als generated in the wake of the discovery of the Jones polynomial. We usually think of this result as applying to knots in the 3-sphere S3. However, it applies equally well to knots in S2×I (where I is the unit interval [0, 1]). Indeed, if one removes two disjoint balls from S3, the resulting space is homeomorphic to S2 × I. It is not hard to see that these two balls do not affect knot equivalence. We conclude that the theory of knot equivalence in S2 × I is the same as in S3. With this equivalence in mind, it is natural to ask if the Tait conjecture generalizes to knots in spaces of the form S × I where S is any compact surface.
    [Show full text]
  • Crosscap Number and Knot Projections
    CROSSCAP NUMBER AND KNOT PROJECTIONS NOBORU ITO AND YUSUKE TAKIMURA Abstract. We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical and new results that determine the set of alternating knots with the crosscap number at most two. 1. Introduction In this paper, we introduce an unknotting-type number of knot projections (Def- inition 1) as follows. Every double point in a knot projection can be spliced two different ways (Figure 2), one of which gives another knot projection (Definition 2). A special case of such operations is a first Reidemeister move RI−, as shown in Fig- ure 2. If the other case of such operations, which is not of type RI−, it is denoted by S−. Beginning with an n-crossing knot projection P , there are many sequences of n splices of type RI− and type S−, all of which end with the simple closed curve O. Then, we define the number u−(P ) as the minimum number of splices of type S− (Definition 3). For this number, we determine the set of knot projections with u−(P ) = 1 or u−(P ) = 2 (Theorem 1, Section 3). Here, we provide an efficient method to obtain a knot projection P with u−(P ) = n for a given n (Move 1). Further, for a connected sum (Definition 4) of knot projections, we show that the additivity of u− under the connected sum (Section 7).
    [Show full text]
  • The Geometric Content of Tait's Conjectures
    The geometric content of Tait's conjectures Ohio State CKVK* seminar Thomas Kindred, University of Nebraska-Lincoln Monday, November 9, 2020 Historical background: Tait's conjectures, Fox's question Tait's conjectures (1898) Let D and D0 be reduced alternating diagrams of a prime knot L. (Prime implies 6 9 T1 T2 ; reduced means 6 9 T .) Then: 0 (1) D and D minimize crossings: j jD = j jD0 = c(L): 0 0 (2) D and D have the same writhe: w(D) = w(D ) = j jD0 − j jD0 : (3) D and D0 are related by flype moves: T 2 T1 T2 T1 Question (Fox, ∼ 1960) What is an alternating knot? Tait's conjectures all remained open until the 1985 discovery of the Jones polynomial. Fox's question remained open until 2017. Historical background: Proofs of Tait's conjectures In 1987, Kauffman, Murasugi, and Thistlethwaite independently proved (1) using the Jones polynomial, whose degree span is j jD , e.g. V (t) = t + t3 − t4. Using the knot signature σ(L), (1) implies (2). In 1993, Menasco-Thistlethwaite proved (3), using geometric techniques and the Jones polynomial. Note: (3) implies (2) and part of (1). They asked if purely geometric proofs exist. The first came in 2017.... Tait's conjectures (1898) T 2 GivenT1 reducedT2 alternatingT1 diagrams D; D0 of a prime knot L: 0 (1) D and D minimize crossings: j jD = j jD0 = c(L): 0 0 (2) D and D have the same writhe: w(D) = w(D ) = j jD0 − j jD0 : (3) D and D0 are related by flype moves: Historical background: geometric proofs Question (Fox, ∼ 1960) What is an alternating knot? Theorem (Greene; Howie, 2017) 3 A knot L ⊂ S is alternating iff it has spanning surfaces F+ and F− s.t.: • Howie: 2(β1(F+) + β1(F−)) = s(F+) − s(F−).
    [Show full text]
  • Crosscap Numbers of Alternating Knots Via Unknotting Splices
    Crosscap numbers of alternating knots via unknotting splices THOMAS KINDRED Ito-Takimura recently defined a splice-unknotting number u−(D) for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes. 57M25 1 Introduction Let K ⊂ S3 be a knot. An embedded, compact, connected surface F ⊂ S3 is said to span K if @F = K . The crosscap number of K , denoted cc(K), is the smallest value of 12 β1(F) among all 1-sided spanning surfaces for K. A theorem of Adams and the author [4] states that, given an alternating diagram D of a knot K, the crosscap number of K is realized by some state surface from D. (Section 2 reviews background.) Moreover, given such D and K, an algorithm in [4] finds a 1-sided state surface F from D with β1(F) = cc(K). Ito-Takimura recently introduced a splice-unknotting number u−(D) for knot diagrams. Minimizing this number across all diagrams of a given knot K defines a knot invariant, arXiv:1905.11367v1 [math.GT] 27 May 2019 u−(K). After proving that u−(D) ≥ cc(K) holds for any diagram D of any nontrivial knot K, Ito-Takimura ask whether this inequality is ever strict in the case of prime alternating diagrams.
    [Show full text]
  • CROSSCAP NUMBERS of PRETZEL KNOTS 1. Introduction It Is Well
    CROSSCAP NUMBERS OF PRETZEL KNOTS 市原 一裕 (KAZUHIRO ICHIHARA) 大阪産業大学教養部 (OSAKA SANGYO UNIVERSITY) 水嶋滋氏 (東京工業大学大学院情報理工学研究科) との共同研究 (JOINT WORK WITH SHIGERU MIZUSHIMA (TOKYO INSTITUTE OF TECHNOLOGY)) Abstract. For a non-trivial knot in the 3-sphere, the crosscap number is de¯ned as the minimal ¯rst betti number of non-orientable spanning surfaces for it. In this article, we report a simple formula of the crosscap number for pretzel knots. 1. Introduction It is well-known that any knot in the 3-sphere S3 bounds an orientable subsurface in S3; called a Seifert surface. One of the most basic invariant in knot theory, the genus of a knot K, is de¯ned to be the minimal genus of a Seifert surface for K. On the other hand, any knot in S3 also bounds a non-orientable subsurface in S3: Consider checkerboard surfaces for a diagram of the knot, one of which is shown to be non-orientable. In view of this, similarly as the genus of a knot, B.E. Clark de¯ned the crosscap number of a knot as follows. De¯nition (Clark, [1]). The crosscap number γ(K) of a knot K is de¯ned to be the minimal ¯rst betti number of a non-orientable surface spanning K in S3. For completeness we de¯ne γ(K) = 0 if and only if K is the unknot. Example. The ¯gure-eight knot K in S3 bounds a once-punctured Klein bottle S, appearing as a checkerboard surface for the diagram of K with minimal crossings. Thus γ(K) · ¯1(S) = 2.
    [Show full text]
  • Peter Guthrie Tait: a Knot's Tale Transcript
    Peter Guthrie Tait: A Knot's Tale Transcript Date: Wednesday, 31 October 2012 - 4:30PM Location: Barnard's Inn Hall 31 October 2012 Peter Guthrie Tait: A Knot's Tale Dr Julia Collins Good afternoon, everyone. Thank you very much for the invitation to come and speak at Gresham College. I have never been here before, so it is really exciting to see so many people. Of the three people that we are talking about this afternoon, I think Peter Guthrie Tait is the one who is least well-known. Put your hand up if you knew before today who Peter Guthrie Tait was… Okay, put your hand down if you are a member of the BSHM… [Laughter] I think, of the three, he is certainly the least well-known, so my job today is to tell you a bit about his life story, and in particular the contribution that he made to the mathematical theory of knots. At this point, I want to say, the caveat to that, I am not a historian and I am not a physicist. I do not understand the physics that Tait did, so I will be talking about his mathematics, and we can talk more about other things during the break. I also apologise to any Tait enthusiasts that there will be so many things about his life that I do not have time to fit into 45 minutes, so I apologise for that. I realise that I am the person standing between you and the alcoholic drinks in 45 minutes, so let me motivate this lecture to make you excited about what is coming up in my talk! [Recording plays] I am going to start the story in Edinburgh, but in modern times, with me, the narrator.
    [Show full text]
  • Knots, Molecules, and the Universe: an Introduction to Topology
    KNOTS, MOLECULES, AND THE UNIVERSE: AN INTRODUCTION TO TOPOLOGY AMERICAN MATHEMATICAL SOCIETY https://doi.org/10.1090//mbk/096 KNOTS, MOLECULES, AND THE UNIVERSE: AN INTRODUCTION TO TOPOLOGY ERICA FLAPAN with Maia Averett David Clark Lew Ludwig Lance Bryant Vesta Coufal Cornelia Van Cott Shea Burns Elizabeth Denne Leonard Van Wyk Jason Callahan Berit Givens Robin Wilson Jorge Calvo McKenzie Lamb Helen Wong Marion Moore Campisi Emille Davie Lawrence Andrea Young AMERICAN MATHEMATICAL SOCIETY 2010 Mathematics Subject Classification. Primary 57M25, 57M15, 92C40, 92E10, 92D20, 94C15. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-96 Library of Congress Cataloging-in-Publication Data Flapan, Erica, 1956– Knots, molecules, and the universe : an introduction to topology / Erica Flapan ; with Maia Averett [and seventeen others]. pages cm Includes index. ISBN 978-1-4704-2535-7 (alk. paper) 1. Topology—Textbooks. 2. Algebraic topology—Textbooks. 3. Knot theory—Textbooks. 4. Geometry—Textbooks. 5. Molecular biology—Textbooks. I. Averett, Maia. II. Title. QA611.F45 2015 514—dc23 2015031576 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service.
    [Show full text]
  • Arxiv:1202.6488V4 [Math.GT] 11 Mar 2015 Ikinvariants Link 2 Introduction 1 Contents Invertibility
    Prime component-preservingly amphicheiral link with odd minimal crossing number Teruhisa KADOKAMI and Yoji KOBATAKE March 11, 2015 Abstract For every odd integer c 21, we raise an example of a prime component-preservingly ≥ amphicheiral link with the minimal crossing number c. The link has two components, and consists of an unknot and a knot which is ( )-amphicheiral with odd minimal crossing − number. We call the latter knot a Stoimenow knot. We also show that the Stoimenow knot is not invertible by the Alexander polynomials. Contents 1 Introduction 1 2 Link invariants 4 2.1 Kauffmanbracket.............................. 4 2.2 AlexanderandConwaypolynomials. 6 3 Stoimenow knots 7 4 Proof of Theorem 1.3 9 arXiv:1202.6488v4 [math.GT] 11 Mar 2015 5 Non-invertibility of Stoimenow knots 13 1 Introduction 3 Let L = K1 Kr be an oriented r-component link in S . A 1-component link is called a knot.∪···∪ For an oriented knot K, we denote the orientation-reversed knot by 3 K. If ϕ is an orientation-reversing homeomorphism of S so that ϕ(Ki) = εσ(i)Kσ(i) for− all i = 1,...,r where ε =+ or , and σ is a permutation of 1, 2,...,r , then L i − { } 2010 Mathematics Subject Classification: 57M25, 57M27. Keywords: component-preservingly amphicheiral link; minimal crossing number; Tait’s conjecture ; invertibility. 1 is called an (ε1,...,εr; σ)-amphicheiral link. A term “amphicheiral link” is used as a general term for an (ε1,...,εr; σ)-amphicheiral link. If ϕ can be taken as an involution (i.e. ϕ2 = id), then L is called a strongly amphicheiral link.
    [Show full text]