DOCSLIB.ORG

Wild Wild Whitehead Danny Calegari

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Wild Wild Whitehead Danny Calegari SHORT STORIES Wild Wild Whitehead Danny Calegari Take a knot. Take a tube around the knot. Put a new knot result can be shrunk down to a point. The track of the knot in the tube, twisted around and clasping itself as in Fig- 퐾1 during this process sweeps out an immersed disk. ure 1. The new knot goes once around the tube, and then In terms of the fundamental group, a knot 퐾 in a space “doubles back” and clasps itself. The new knot is the White- 푋 determines a conjugacy class [퐾] in the fundamental head double of the old. group 휋1(푋). Now, 휋1(푁0) = ℤ, and since 퐾1 bounds an immersed disk, [퐾1] is trivial in 휋1(푁0). We can keep going. Let 푁1 be a tube around 퐾1, thin enough to fit in 푁0, and let 퐾2 ⊂ 푁1 be the Whitehead double of 퐾1. And so on. Each 퐾푖 bounds an embedded disk in 푆3, but each of these disks must go (many times!) all the way outside 푁0. The tubes get thinner and thinner as we go, and longer Figure 1. The green knot is the Whitehead double of the black and longer. Consequently, the knots must get longer and knot. longer too: Each 퐾푛 must wind back and forth at least 푛 2 times around 푁0, clasping itself in a complicated way at the end. The infinite intersection ⋂ 푁푖 is called the Let’s call the first knot 퐾 and its Whitehead double 퐾 , 푖 0 1 Whitehead continuum, which we write Wh, see Figure 2. The and let’s call the tube around 퐾 (actually a solid torus) 0 Whitehead continuum is connected but not path connected. 푁 . It makes sense to take the Whitehead double of any 0 It has an entangled dyadic Cantor set of “strands” that wind knot, but in Figure 1, 퐾0 is a trivial knot; i.e., it bounds 3 around 푁0. an embedded disk in 푆 . In this case, 퐾1 is trivial too: it 3 3 The complement 푆 −Wh is an open 3-manifold called bounds an embedded disk in 푆 . But 퐾 is knotted in 푁 . 3 1 0 the Whitehead manifold. It turns out that 푆 − Wh is con- Any embedded disk that 퐾 bounds must go outside 푁 . 1 0 tractible but not homeomorphic to a 3-ball. Let’s see why. On the other hand, 퐾1 is homotopically trivial in 푁0; i.e., 3 ′ The outside of 푁0 in 푆 is another solid torus 푁0, whose it bounds an immersed disk, one that crosses itself, but does ′ ′ core is a knot 퐾0 linking 퐾0 in a Hopf link. [퐾0] is the not cross 푁0. To see this, just push one of the clasps of 퐾1 3 ′ generator of 휋1(푆 − 푁0) = 휋1(푁0) = ℤ. The knots through the other one; this undoes the knotting, and the ′ 퐾0 and 퐾1 together form a 2-component link called the Whitehead link. This link is symmetric: we can isotope it Danny Calegari is a professor of mathematics at the University of Chicago. His 퐾′ 퐾 email address is [email protected]. around and interchange 0 and 1; see Figure 3. 3 ′ For permission to reprint this article, please contact: Since [퐾1] is trivial in 휋1(푁0) = 휋1(푆 − 푁0), it ′ 3 [email protected]. follows by symmetry that [퐾0] is trivial in 휋1(푆 − 푁1). 3 3 DOI: https://doi.org/10.1090/noti1837 Consequently the inclusion map 푆 − 푁0 → 푆 − 푁1 APRIL 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 581 Short Stories Figure 2. Successive approximations converge to Wh. Nudging adjusts points in 푁1 ×ℝ by sliding each point ×ℝ slightly backward or forward in time. Nudging ex- tends to a self-homeomorphism 휈 of 푁0 × ℝ, fixed on the boundary. Figure 3. The Whitehead link is symmetric. induces the zero map on 휋1. Each 푁푖 is unknotted in 3 푆 , and each 푁푖+1 sits in 푁푖 the same way that 푁1 sits 3 in 푁0. So each 휋1(푆 − 푁푖) = ℤ, and each inclusion 3 3 푆 − 푁푛 → 푆 − 푁푛+1 induces the zero map on 휋1. Tak- 휋 (푆3 − ) ing a direct limit, 1 Wh itself is trivial. A similar Figure 4. Folding the clasps of each 푁1 back and forth in time 3 argument shows that all the homotopy groups of 푆 −Wh nudges the union of all 푁1’s into a collection of spirals. vanish, and it is contractible. 푆3 − − 푁′ = On the other hand, the complement Wh 0 By the way, there’s not just one spiral—there’s a circle’s 푁0 − Wh has an infinitely generated fundamental group; worth of them, filling the whole of 푁1 × ℝ. Two slices each 휋1(푁푖−푁푖+1) is complicated (it contains free groups 휈(푁1 × 푡), 휈(푁1 × 푠) are in the same spiral if and only if of every rank), and all of them include as subgroups of 푡 − 푠 휖/2 3 is an integer multiple of . 휋1(푁0 − Wh). This shows that 푆 − Wh is not a ball. 3 After nudging, the next move will straighten out this Now let’s compactify 푆 − Wh by adding a single point and every other spiral so that its projection to the 푆3 factor at infinity. This compact space can also be thought ofas < 휖/2 3 is small (let’s say for concreteness it has diameter ) the quotient space 푆 /Wh that we get by crushing Wh to a without affecting the projection to the ℝ factor. 3 3 single point. Because 푆 − Wh is not a ball, 푆 /Wh is not 3 The cylinder 퐾0 × ℝ ⊂ 푆 × ℝ has polar coordinates a manifold. However—remarkably—it is a manifold factor: (휃, 푡) where 휃 ∈ ℝ/ℤ. Extend these polar coordinates to a (푆3/ ) × ℝ 푆3 × ℝ the product Wh is homeomorphic to ! small tubular neighborhood of 퐾0 ×ℝ containing 푁1 ×ℝ, How can this possibly be??? with closure contained in the interior of 푁0 × ℝ. First, each 푁1 slice can be unknotted by a tiny perturba- We can “untwist” every spiral simultaneously by the map tion in 푁0 × ℝ. To distinguish the ℝ factor and, for the sake of brevity, we refer to it as the “time” coordinate (this (휃, 푡) → (휃 − 2푡/휖, 푡) is purely a notational convenience). In this language, we unclasp 푁1 from itself by nudging one clasp very slightly on our small tubular neighborhood. Twisting extends to forward into the future, and the other very slightly back a self-homeomorphism 휏 of 푁0 × ℝ, once again fixed on into the past. After the nudge, 푁1 will not clasp itself, but the boundary. it will clasp a “future” 푁1 on one side, and a “past” 푁1 In summary, first we nudge, then we twist. After doing on the other. Instead of 푁1 clasping itself in a circle, we this, every 휏휈(푁1) slice projects to subsets of diameter at 3 get a chain of successive 푁1’s, each clasping the next, in a most 휖/2 in both the ℝ and the 푆 directions. So 휏휈(푁1) slowly ascending spiral. Let’s let 휖/4 be the size of the per- has diameter at most 휖. turbations of each clasp in the time direction, so that the In other words, ℎ1 ∶= 휏휈 simultaneously shrinks all 3 projection of each 푁1 to the time coordinate after it’s been the 푁1 slices in 푆 × ℝ as small as we like, while keeping 3 nudged has total length 휖/2. (푆 − 푁0) × ℝ fixed pointwise. 582 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4 GET MORE FROM YOUR JOURNALS Learn about AMS MathViewer: n interactive, dual-panel reading experience Figure 5. Screw the top and bottom in opposite directions like for journals published by the American you’re taking the lid off a pickle jar. A Mathematical Society. Take a sequence 휖푖 → 0, and repeat this operation for AMS MathViewer’s responsive HTML format each 푖 > 1 in place of 1 with 휖푖 in place of 휖. We get a se- perfectly renders mathematical content on 3 3 desktop, tablet, or phone. Click on embedded quence of self-homeomorphisms ℎ푖 ∶ 푆 × ℝ → 푆 × links to formulas, theorems, gures, and references ℝ, each supported in 푁 × ℝ, as a composition of a 푖−1 to view them in the second panel, in tandem with nudge-and-twist ℎ푖 ∶= 휏푖휈푖. Each 푁푖 slice gets smaller the original text. and smaller in diameter as we apply consecutive ℎ푖’s. Un- der application of successive ℎ푖’s, the orbit of every point is a Cauchy sequence, and the infinite composition 3 3 ℎ ∶= lim ℎ푖 ℎ푖−1 ⋯ ℎ1 ∶ 푆 × ℝ → 푆 × ℝ 푖→∞ is well-defined and continuous. For any compact subset 푋 of 푆3 − Wh the restriction of ℎ to 푋 is the composition of finitely many homeomor- phisms, so the restriction of ℎ to (푆3 −Wh)×ℝ is a home- omorphism. On the other hand, each Wh slice is succes- sively shrunk smaller and smaller by successive ℎ푖, so in the end ℎ crushes each Wh slice to a point, and ℎ factors as ℎ = 푔휋, 휋 푔 푆3 × ℝ −→(푆3/Wh) × ℝ −→푆3 × ℝ, where 휋 ∶ 푆3 × ℝ → (푆3/Wh) × ℝ is the quotient map, and 푔 is the homeomorphism we’ve been looking for. AUTHOR’S NOTE. The main theorem in this article CURRENTLY AVAILABLE FOR: and its proof are both well-known, and not due to me! Transactions of the AMS—Series B They are due to J.
Load more

Recommended publications
  • Knots: a Handout for Mathcircles
    Knots: a handout for mathcircles Mladen Bestvina February 2003 1 Knots Informally, a knot is a knotted loop of string. You can create one easily enough in one of the following ways: • Take an extension cord, tie a knot in it, and then plug one end into the other. • Let your cat play with a ball of yarn for a while. Then find the two ends (good luck!) and tie them together. This is usually a very complicated knot. • Draw a diagram such as those pictured below. Such a diagram is a called a knot diagram or a knot projection. Trefoil and the figure 8 knot 1 The above two knots are the world's simplest knots. At the end of the handout you can see many more pictures of knots (from Robert Scharein's web site). The same picture contains many links as well. A link consists of several loops of string. Some links are so famous that they have names. For 2 2 3 example, 21 is the Hopf link, 51 is the Whitehead link, and 62 are the Bor- romean rings. They have the feature that individual strings (or components in mathematical parlance) are untangled (or unknotted) but you can't pull the strings apart without cutting. A bit of terminology: A crossing is a place where the knot crosses itself. The first number in knot's \name" is the number of crossings. Can you figure out the meaning of the other number(s)? 2 Reidemeister moves There are many knot diagrams representing the same knot. For example, both diagrams below represent the unknot.
    [Show full text]
  • Dehn Filling of the “Magic” 3-Manifold
    communications in analysis and geometry Volume 14, Number 5, 969–1026, 2006 Dehn filling of the “magic” 3-manifold Bruno Martelli and Carlo Petronio We classify all the non-hyperbolic Dehn fillings of the complement of the chain link with three components, conjectured to be the smallest hyperbolic 3-manifold with three cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many one-cusped and two-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: • for every integer n, we can prove that there are infinitely many hyperbolic knots in S3 having exceptional surgeries {n, n +1, n +2,n+3}, with n +1,n+ 2 giving small Seifert manifolds and n, n + 3 giving toroidal manifolds. • we exhibit a two-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements. • we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements. • we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling. 0. Introduction We study in this paper the Dehn fillings of the complement N of the chain link with three components in S3, shown in figure 1. The hyperbolic structure of N was first constructed by Thurston in his notes [28], and it was also noted there that the volume of N is particularly small. The relevance of N to three-dimensional topology comes from the fact that by filling N, one gets most of the hyperbolic manifolds known and most of the interesting non-hyperbolic fillings of cusped hyperbolic manifolds.
    [Show full text]
  • Hyperbolic Structures from Link Diagrams
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2012 Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Geometry and Topology Commons Recommended Citation Tsvietkova, Anastasiia, "Hyperbolic Structures from Link Diagrams. " PhD diss., University of Tennessee, 2012. https://trace.tennessee.edu/utk_graddiss/1361 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Anastasiia Tsvietkova entitled "Hyperbolic Structures from Link Diagrams." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Morwen B. Thistlethwaite, Major Professor We have read this dissertation and recommend its acceptance: Conrad P. Plaut, James Conant, Michael Berry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Hyperbolic Structures from Link Diagrams A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Anastasiia Tsvietkova May 2012 Copyright ©2012 by Anastasiia Tsvietkova. All rights reserved. ii Acknowledgements I am deeply thankful to Morwen Thistlethwaite, whose thoughtful guidance and generous advice made this research possible.
    [Show full text]
  • 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 the Kinematic Formula in the Lives of the Saints Danny Calegari
    SHORT STORIES On the Kinematic Formula in the Lives of the Saints Danny Calegari Saint Sebastian was an early Christian martyr. He served as a captain in the Praetorian Guard under Diocletian un- til his religious faith was discovered, at which point he was taken to a field, bound to a stake, and shot by archers “till he was as full of arrows as an urchin1 is full of pricks2.” Rather miraculously, he made a full recovery, but was later executed anyway for insulting the emperor. The trans- pierced saint became a popular subject for Renaissance painters, e.g., Figure 1. The arrows in Mantegna’s painting have apparently ar- rived from all directions, though they are conspicuously grouped around the legs and groin, almost completely missing the thorax. Intuitively, we should expect more ar- rows in the parts of the body that present a bigger cross section. This intuition is formalized by the claim that a subset of the surface of Saint Sebastian has an area pro- portional to its expected number of intersections with a random line (i.e., arrow). Since both area and expectation are additive, we may reduce the claim (by polygonal ap- proximation and limit) to the case of a flat triangular Saint Sebastian, in which case it is obvious. Danny Calegari is a professor of mathematics at the University of Chicago. His Figure 1. Andrea Mantegna’s painting of Saint Sebastian. email address is [email protected]. 1 hedgehog This is a 3-dimensional version of the classical Crofton 2quills formula, which says that the length of a plane curve is pro- For permission to reprint this article, please contact: portional to its expected number of intersections with a [email protected].
    [Show full text]
  • Groups of PL Homeomorphisms of Cubes
    GROUPS OF PL HOMEOMORPHISMS OF CUBES DANNY CALEGARI AND DALE ROLFSEN D´edi´e`aMichel Boileau sur son soixanti`eme anniversaire. Sant´e! Abstract. We study algebraic properties of groups of PL or smooth homeo- morphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing point- wise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted. 1. Introduction We are concerned in this paper with algebraic properties of the group of PL homeomorphisms of a PL manifold, fixed on some PL submanifold (usually of codimension 1 or 2, for instance the boundary) and some of its subgroups (usually those preserving some structure). The most important case is the group of PL homeomorphisms of In fixed pointwise on ∂In; hence these are “groups of PL homeomorphisms of the (n-)cube”. The algebraic study of transformation groups (often in low dimension, or preserv- ing some extra structure such as a symplectic or complex structure) has recently seen a lot of activity; however, much of this activity has been confined to the smooth category. It is striking that many of these results can be transplanted to the PL category. This interest is further strengthened by the possibility of working in the PL category over (real) algebraic rings or fields (this possibility has already been exploited in dimension 1, in the groups F and T of Richard Thompson).
    [Show full text]
  • Commentary on Thurston's Work on Foliations
    COMMENTARY ON FOLIATIONS* Quoting Thurston's definition of foliation [F11]. \Given a large supply of some sort of fabric, what kinds of manifolds can be made from it, in a way that the patterns match up along the seams? This is a very general question, which has been studied by diverse means in differential topology and differential geometry. ... A foliation is a manifold made out of striped fabric - with infintely thin stripes, having no space between them. The complete stripes, or leaves, of the foliation are submanifolds; if the leaves have codimension k, the foliation is called a codimension k foliation. In order that a manifold admit a codimension- k foliation, it must have a plane field of dimension (n − k)." Such a foliation is called an (n − k)-dimensional foliation. The first definitive result in the subject, the so called Frobenius integrability theorem [Fr], concerns a necessary and sufficient condition for a plane field to be the tangent field of a foliation. See [Spi] Chapter 6 for a modern treatment. As Frobenius himself notes [Sa], a first proof was given by Deahna [De]. While this work was published in 1840, it took another hundred years before a geometric/topological theory of foliations was introduced. This was pioneered by Ehresmann and Reeb in a series of Comptes Rendus papers starting with [ER] that was quickly followed by Reeb's foundational 1948 thesis [Re1]. See Haefliger [Ha4] for a detailed account of developments in this period. Reeb [Re1] himself notes that the 1-dimensional theory had already undergone considerable development through the work of Poincare [P], Bendixson [Be], Kaplan [Ka] and others.
    [Show full text]
  • A New Technique for the Link Slice Problem
    Invent. math. 80, 453 465 (1985) /?/ven~lOnSS mathematicae Springer-Verlag 1985 A new technique for the link slice problem Michael H. Freedman* University of California, San Diego, La Jolla, CA 92093, USA The conjectures that the 4-dimensional surgery theorem and 5-dimensional s-cobordism theorem hold without fundamental group restriction in the to- pological category are equivalent to assertions that certain "atomic" links are slice. This has been reported in [CF, F2, F4 and FQ]. The slices must be topologically flat and obey some side conditions. For surgery the condition is: ~a(S 3- ~ slice)--, rq (B 4- slice) must be an epimorphism, i.e., the slice should be "homotopically ribbon"; for the s-cobordism theorem the slice restricted to a certain trivial sublink must be standard. There is some choice about what the atomic links are; the current favorites are built from the simple "Hopf link" by a great deal of Bing doubling and just a little Whitehead doubling. A link typical of those atomic for surgery is illustrated in Fig. 1. (Links atomic for both s-cobordism and surgery are slightly less symmetrical.) There has been considerable interplay between the link theory and the equivalent abstract questions. The link theory has been of two sorts: algebraic invariants of finite links and the limiting geometry of infinitely iterated links. Our object here is to solve a class of free-group surgery problems, specifically, to construct certain slices for the class of links ~ where D(L)eCg if and only if D(L) is an untwisted Whitehead double of a boundary link L.
    [Show full text]
  • Universal Circles for Quasigeodesic Flows 1 Introduction
    Geometry & Topology 10 (2006) 2271–2298 2271 arXiv version: fonts, pagination and layout may vary from GT published version Universal circles for quasigeodesic flows DANNY CALEGARI We show that if M is a hyperbolic 3–manifold which admits a quasigeodesic flow, then π1(M) acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic 3–manifold without a quasigeodesic flow, answering a long-standing question of Thurston. 57R30; 37C10, 37D40, 53C23, 57M50 1 Introduction 1.1 Motivation and background Hyperbolic 3–manifolds can be studied from many different perspectives. A very fruitful perspective is to think of such manifolds as dynamical objects. For example, a very important class of hyperbolic 3–manifolds are those arising as mapping tori of pseudo-Anosov automorphisms of surfaces (Thurston [24]). Such mapping tori naturally come with a flow, the suspension flow of the automorphism. In the seminal paper [4], Cannon and Thurston showed that this suspension flow can be chosen to be quasigeodesic and pseudo-Anosov. Informally, a flow on a hyperbolic 3–manifold is quasigeodesic if the lifted flowlines in the universal cover are quasi-geodesics in H3 , and a flow is pseudo-Anosov if it looks locally like a semi-branched cover of an Anosov flow. Of course, such a flow need not be transverse to a foliation by surfaces. See Thurston [24] and Fenley [6] for background and definitions.
    [Show full text]
  • UW Math Circle May 26Th, 2016
    UW Math Circle May 26th, 2016 We think of a knot (or link) as a piece of string (or multiple pieces of string) that we can stretch and move around in space{ we just aren't allowed to cut the string. We draw a knot on piece of paper by arranging it so that there are two strands at every crossing and by indicating which strand is above the other. We say two knots are equivalent if we can arrange them so that they are the same. 1. Which of these knots do you think are equivalent? Some of these have names: the first is the unkot, the next is the trefoil, and the third is the figure eight knot. 2. Find a way to determine all the knots that have just one crossing when you draw them in the plane. Show that all of them are equivalent to an unknotted circle. The Reidemeister moves are operations we can do on a diagram of a knot to get a diagram of an equivalent knot. In fact, you can get every equivalent digram by doing Reidemeister moves, and by moving the strands around without changing the crossings. Here are the Reidemeister moves (we also include the mirror images of these moves). We want to have a way to distinguish knots and links from one another, so we want to extract some information from a digram that doesn't change when we do Reidemeister moves. Say that a crossing is postively oriented if it you can rotate it so it looks like the left hand picture, and negatively oriented if you can rotate it so it looks like the right hand picture (this depends on the orientation you give the knot/link!) For a link with two components, define the linking number to be the absolute value of the number of positively oriented crossings between the two different components of the link minus the number of negatively oriented crossings between the two different components, #positive crossings − #negative crossings divided by two.
    [Show full text]
  • The Kauffman Bracket and Genus of Alternating Links
    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 6-2016 The Kauffman Bracket and Genus of Alternating Links Bryan M. Nguyen Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Other Mathematics Commons Recommended Citation Nguyen, Bryan M., "The Kauffman Bracket and Genus of Alternating Links" (2016). Electronic Theses, Projects, and Dissertations. 360. https://scholarworks.lib.csusb.edu/etd/360 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. The Kauffman Bracket and Genus of Alternating Links A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Bryan Minh Nhut Nguyen June 2016 The Kauffman Bracket and Genus of Alternating Links A Thesis Presented to the Faculty of California State University, San Bernardino by Bryan Minh Nhut Nguyen June 2016 Approved by: Dr. Rolland Trapp, Committee Chair Date Dr. Gary Griffing, Committee Member Dr. Jeremy Aikin, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract Giving a knot, there are three rules to help us finding the Kauffman bracket polynomial. Choosing knot's orientation, then applying the Seifert algorithm to find the Euler characteristic and genus of its surface. Finally finding the relationship of the Kauffman bracket polynomial and the genus of the alternating links is the main goal of this paper.
    [Show full text]
  • Department of Mathematics 1
    Department of Mathematics 1 Department of Mathematics Chair • Shmuel Weinberger Professors • Laszlo Babai, Computer Science and Mathematics • Guillaume Bal, Statistics and Mathematics • Alexander A. Beilinson • Danny Calegari • Francesco Calegari • Kevin Corlette • Marianna Csörnyei • Vladimir Drinfeld • Todd Dupont, Computer Science and Mathematics • Matthew Emerton • Alex Eskin • Benson Farb • Robert A. Fefferman • Victor Ginzburg • Denis Hirschfeldt • Kazuya Kato • Carlos E. Kenig • Gregory Lawler, Mathematics and Statistics • Maryanthe Malliaris • J. Peter May • Andre Neves • Bao Châu Ngô • Madhav Vithal Nori • Alexander Razborov, Mathematics and Computer Science • Luis Silvestre • Charles Smart • Panagiotis Souganidis • Sidney Webster • Shmuel Weinberger • Amie Wilkinson • Robert Zimmer Associate Professors • Simion Filip • Ewain Gwynne Assistant Professors • Sebastian Hurtado-Salazar • Dana Mendelson • Nikita Rozenblyum • Daniil Rudenko Instructors • Lucas Benigni • Guher Camliyurt • Stephen Cantrell • Elliot Cartee • Mark Cerenzia 2 Department of Mathematics • Andrea Dotto • Mikolaj Fraczyk • Pedro Gasper • Kornelia Hera • Trevor Hyde • Kasia Jankiewicz • Justin Lanier • Brian Lawrence • Zhilin Luo • Akhil Mathew • Henrik Matthieson • Cornelia Mihaila • Lucia Mocz • Benedict Morrissey • Davi Obata • Lue Pan • Wenyu Pan • Beniada Shabani • Danny Shi • Daniel Stern • Ao Sun • Xuan Wu • Zihui Zhao • Jinping Zhuge Senior Lecturers • John Boller • Lucas Culler • Jitka Stehnova • Sarah Ziesler Lecturer • Meghan Anderson Assistant Instructional
    [Show full text]