DOCSLIB.ORG

Links Not Concordant to the Hopf Link

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Links Not Concordant to the Hopf Link Links Not Concordant to the Hopf Link Mark Powell, Indiana University Links Not Concordant to the Hopf Link Mark Powell, Indiana University 12th March 2012 1 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Joint work with Stefan Friedl. 2 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University The Hopf link: 3 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Definition Two n-component links L; L0 : S1 t S1 · · · t S1 ⊆ S3 are concordant if they cobound n embedded annuli G S1 × I ⊂ S3 × I n with boundary L ⊂ S3 × f0g and L0 ⊂ S3 × f1g. CAT = TOP, all embeddings are to be locally flat. 4 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Aim: given a 2-component link, find algebraic obstructions to it being concordant to the Hopf link H. Find examples with unknotted components and so that previously known obstructions vanish. We produce an obstruction theory in the spirit of the Casson{Gordon knot slicing obstructions. 5 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Lemma Any link which is concordant to a given 2 component link L has the same linking number as L. 6 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Link exterior: 3 XL := cl(S n νL): Using ' 2 π1(XL) ! H1(XL) −! Z ; define the multi{variable Alexander polynomial corresponding to the universal abelian cover: ±1 ±1 ∆L(s; t) := ord(H1(XL; Z[t ; s ])) Note that ∆H (s; t) = 1. 7 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Theorem (Kawauchi) If a 2 component link with linking number 1 is concordant to H then −1 −1 ∆L(s; t) ≡ f (t; s)f (t ; s ); up to ±tk sl . 8 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Partial converse generalising Freedman: Theorem (J. Davis 2004) A 2 component link with multivariable Alexander polynomial 1 is topologically concordant to the Hopf link. 9 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University I Cha, Kim, Ruberman and Strle showed that this result does not hold in the smooth category. I One imagines concordance to H admits a filtration analogous to the Cochran-Harvey-Horn positive and negative filtrations of knot concordance. 10 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University We form a closed 3 manifold ML := XL [ XH ' ∼ 1 1 1 1 gluing along @XL −! @XH = S × S t S × S . ∼ 1 1 1 Note that MH = S × S × S . 11 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Pick a prime p, a; b 2 N and a homomorphism to a p-group: ': H1(ML) ! A = Zpa ⊕ Zpb a+b ' The maps '; 'j induce finite p -fold coverings ML and ' XL . 12 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University We define Ya; Yb ⊂ @XL as in the diagram. Ya Ya XL Yb Yb 13 / 43 ' ' Links Not Recall that X is the p-primary cover induced by '. Ya is Concordant to the L Hopf Link the induced (disconnected) cover. Mark Powell, Indiana University Lemma (Friedl,P) There is a non-singular linking form ' ' ' ' TH1(XL ; Ya ) × TH1(XL ; Ya ) ! Q=Z: ∼ F 1 If L is concordant to H with concordance C = 2 S × I, and we define the exterior of the concordance 3 EC := cl(S × I n νC) then there is a metaboliser P = P? for the linking form given by ' ' ' ' P := ker TH1(XL ; Ya ) ! TH1(EC ; Ya ) : 14 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Definition A Seifert surface F ⊂ S3 is a compact, connected, orientable 3 surface embedded in S with @F = K. For x; y 2 H1(F ), define the Seifert form + V : H1(F ) × H1(F ) ! Z; V (x; y) := lk(x; y ); with y + a positive push off of y along the normal to F . 15 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University 1 For ! 2 S ⊂ C, the Levine-Tristram signature σK (!) of K at ! is the average of the one sided limits of the signature function: T σV (!) = σ (1 − !)V + (1 − !)V : where V is a Seifert form for K. These signatures are concordance invariants, and appear in the calculation of our obstructions. 16 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University A knot is Algebraically Null Concordant if there is a Seifert surface F and a basis of H1(F ; Z) with respect to which the Seifert matrix takes the form: 0 B CD 17 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Casson and Gordon, in 1975, defined obstructions which detected the first knots which were found to be algebraically null concordant, but not concordant to the unknot. We extend their methods to 2-component links with linking number 1. 18 / 43 Links Not Concordant to the Hopf Link Definition Mark Powell, 0 Indiana University The Witt group L (F) of a field, characteristic 6= 2, with involution F is given by stable congruence classes of non-singular Hermitian forms on F vector spaces: [(N; λ)], with addition: [(N; λ)] + [(M; η)] = [(N ⊕ M; λ ⊕ η)]; inverses −[(N; λ)] = [(N; −λ)]; and zero 0 1 ⊕ ; : : F F 1 0 In our case F = C(F ), where F is free abelian of rank 3. 19 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Choose homomorphisms: ' ' ' ∼ 3 φ: H1(ML ) ! H1(ML )=TH1(ML ) ! F = Z ' χ: H1(ML ) ! Zq with gcd(p; q) = 1. We will define a Casson-Gordon style Witt group obstruction ' 0 τ(ML ; φ, χ) 2 L (C(F )) ⊗ Q: 20 / 43 Links Not Concordant to the Main Theorem (Friedl, P) Hopf Link Suppose that L is concordant to H. For all Mark Powell, Indiana University ': H1(ML) ! A; there exists a metaboliser ? ' ' P = P ⊆ TH1(XL ; Ya ) for the torsion linking form, such that for all ' χ: H1(ML ) ! Zq which satisfy that χj ' factors as H1(XL ) ' ' ' δ H1(XL ) ! H1(XL ; Ya ) −! Zq; and that δ(P) = f0g; we have ' τ(ML ; φ, χ) = 0: 21 / 43 Links Not Concordant to the Example time. Temporarily, let L be: Hopf Link Mark Powell, Indiana University I ∆L(s; t) = (ts + 1 − s)(ts + 1 − t)(6= 1). I L is concordant to H. I Linking number is 1. I Components are unknotted. 22 / 43 Links Not Concordant to the Hopf Link Mark Powell, ' Indiana University Idea: Take a satellite of L so that τ(ML ; φ, χ) becomes 3 non-zero. Choose a curve g ⊂ XL which is unknotted in S , and a knot K. g K We obtain the satellite link S := S(L; K; g). 23 / 43 Links Not Concordant to the Hopf Link 0 ' Mark Powell, Taking the signature: σ : L (C) −! Z. Indiana University 1 For ζi 2 S ⊂ C: ' 0 τ(ML ; φ, χ)(ζ1; ζ2; ζ3) 2 L (C) almost everywhere. Define a homomorphism: 0 σC(F ) : L (C(F )) ! R; Z σC(F )(τ(s; t; u)) := σ τ(ζ1; ζ2; ζ3) : S1×S1×S1 24 / 43 Links Not Concordant to the Hopf Link Mark Powell, Results of Litherland (1984) on Casson-Gordon invariants of Indiana University satellite knots extend easily to our setting. Theorem ' ' X σC(F )(τ(MS ; φ, χ)) = σC(F )(τ(ML ; φ, χ))+ σK (!ij ) 2 R; i;j i j where !ij := χ(x y · ge), x; y are generators of A = Zpa ⊕ Zpb and σK (!) is the Levine-Tristram signature of K at !. 25 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University I Pick companion knot K with all positive/negative signatures (e.g. trefoil) I Pick ': H1(XL; Z) = Z ⊕ Z ! Z2 ⊕ Z2 = A. I Pick axis curve g which maps non{zero under χ. ' Then σC(F )(τ(S ; φ, χ)) 6= 0, so S is not concordant to H. Note components of S are still unknotted. 26 / 43 Links Not Maple: Concordant to the Hopf Link ' ' ∼ TH1(XL ; Ya ) = Z9 Mark Powell, Indiana University Also used Maple to identify the curve g as shown below as a curve which lifts to a generator. g This completes the description of an example. 27 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University Jae Choon Cha has generalised the idea, to produce COT/CHL style obstructions. Theorem (J.C.Cha) There are links with two unknotted components which are height n Whitney tower/grope concordant to the Hopf link but not height n:5 Whitney tower/grope concordant to H. 28 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University We made a conjecture, which Jae Choon informs us has been confirmed by a graduate student of his: Theorem (Min Hoon Kim) Let L be a two component link which is height 3.5 Whitney tower/grope concordant to H. Then the conclusion of our main obstruction theorem is satisfied. 29 / 43 Links Not Concordant to the Hopf Link Mark Powell, Indiana University It may well be possible to also show that our link is not concordant to H by blowing down along one component and using Casson-Gordon knot slice obstructions; limitations of satellite construction. However this trick however does not tell us anything about height 3.5 Whitney tower/grope concordance.
Load more

Recommended publications
  • A Symmetry Motivated Link Table
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 August 2018 doi:10.20944/preprints201808.0265.v1 Peer-reviewed version available at Symmetry 2018, 10, 604; doi:10.3390/sym10110604 Article A Symmetry Motivated Link Table Shawn Witte1, Michelle Flanner2 and Mariel Vazquez1,2 1 UC Davis Mathematics 2 UC Davis Microbiology and Molecular Genetics * Correspondence: [email protected] Abstract: Proper identification of oriented knots and 2-component links requires a precise link 1 nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a 2 nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing 3 a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible 4 symmetry types for each topology. The study revisits the methods previously used to disambiguate 5 chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo 6 simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two 7 prime 2-component links with up to nine crossings. Guided by geometrical data, linking number and 8 the symmetry groups of 2-component links, a canonical link diagram for each link type is proposed. 9 2 2 2 2 2 2 All diagrams but six were unambiguously chosen (815, 95, 934, 935, 939, and 941). We include complete 10 tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also 11 prove a result on the behavior of the writhe under local lattice moves.
    [Show full text]
  • 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]
  • An Introduction to Knot Theory and the Knot Group
    AN INTRODUCTION TO KNOT THEORY AND THE KNOT GROUP LARSEN LINOV Abstract. This paper for the University of Chicago Math REU is an expos- itory introduction to knot theory. In the first section, definitions are given for knots and for fundamental concepts and examples in knot theory, and motivation is given for the second section. The second section applies the fun- damental group from algebraic topology to knots as a means to approach the basic problem of knot theory, and several important examples are given as well as a general method of computation for knot diagrams. This paper assumes knowledge in basic algebraic and general topology as well as group theory. Contents 1. Knots and Links 1 1.1. Examples of Knots 2 1.2. Links 3 1.3. Knot Invariants 4 2. Knot Groups and the Wirtinger Presentation 5 2.1. Preliminary Examples 5 2.2. The Wirtinger Presentation 6 2.3. Knot Groups for Torus Knots 9 Acknowledgements 10 References 10 1. Knots and Links We open with a definition: Definition 1.1. A knot is an embedding of the circle S1 in R3. The intuitive meaning behind a knot can be directly discerned from its name, as can the motivation for the concept. A mathematical knot is just like a knot of string in the real world, except that it has no thickness, is fixed in space, and most importantly forms a closed loop, without any loose ends. For mathematical con- venience, R3 in the definition is often replaced with its one-point compactification S3. Of course, knots in the real world are not fixed in space, and there is no interesting difference between, say, two knots that differ only by a translation.
    [Show full text]
  • Categorified Invariants and the Braid Group
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 7, July 2015, Pages 2801–2814 S 0002-9939(2015)12482-3 Article electronically published on February 26, 2015 CATEGORIFIED INVARIANTS AND THE BRAID GROUP JOHN A. BALDWIN AND J. ELISENDA GRIGSBY (Communicated by Daniel Ruberman) Abstract. We investigate two “categorified” braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer ho- mology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group. In particular, our proof in the Khovanov case is completely combinatorial. 1. Introduction Recall that the n-strand braid group Bn admits the presentation σiσj = σj σi if |i − j|≥2, Bn = σ1,...,σn−1 , σiσj σi = σjσiσj if |i − j| =1 where σi corresponds to a positive half twist between the ith and (i + 1)st strands. Given a word w in the generators σ1,...,σn−1 and their inverses, we will denote by σ(w) the corresponding braid in Bn. Also, we will write σ ∼ σ if σ and σ are conjugate elements of Bn. As with any group described in terms of generators and relations, it is natural to look for combinatorial solutions to the word and conjugacy problems for the braid group: (1) Word problem: Given words w, w as above, is σ(w)=σ(w)? (2) Conjugacy problem: Given words w, w as above, is σ(w) ∼ σ(w)? The fastest known algorithms for solving Problems (1) and (2) exploit the Gar- side structure(s) of the braid group (cf.
    [Show full text]
  • Knot Theory and the Alexander Polynomial
    Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ...............................
    [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]
  • 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]
  • Using Link Invariants to Determine Shapes for Links
    USING LINK INVARIANTS TO DETERMINE SHAPES FOR LINKS DAN TATING Abstract. In the tables of two component links up to nine cross- ing there are 92 prime links. These different links take a variety of forms and, inspired by a proof that Borromean circles are im- possible, the questions are raised: Is there a possibility for the components of links to be geometric shapes? How can we deter- mine if a link can be formed by a shape? Is there a link invariant we can use for this determination? These questions are answered with proofs along with a tabulation of the link invariants; Conway polynomial, linking number, and enhanced linking number, in the following report on “Using Link Invariants to Determine Shapes for Links”. 1. An Introduction to Links By definition, a link is a set of knotted loops all tangled together. Two links are equivalent if we can deform the one link to the other link without ever having any one of the loops intersect itself or any of the other loops in the process [1].We tabulate links by using projections that minimize the number of crossings. Some basic links are shown below. Notice how each of these links has two loops, or components. Al- though links can have any finite number of components, we will focus This research was conducted as part of a 2003 REU at CSU, Chico supported by the MAA’s program for Strengthening Underrepresented Minority Mathematics Achievement and with funding from the NSF and NSA. 1 2 DAN TATING on links of two and three components.
    [Show full text]
  • I.2 Curves and Knots
    8 I Graphs I.2 Curves and Knots Topology gets quickly more complicated when the dimension increases. The first step up from (0-dimensional) points are (1-dimensional) curves. Classify- ing them intrinsically is straightforward but can be complicated extrinsically. Curve models. A homeomorphism between two topological spaces is a con- tinuous bijection h : X ! Y whose inverse is also continuous. If such a map exists then X and Y are considered the same. More formally, X and Y are said to be homeomorphic or topology equivalent. If two spaces are homeo- morphic then they are either both connected or both not connected. Any two closed intervals are homeomorphic so it makes sense to use the unit in- terval, [0; 1], as the standard model. Similarly, any two half-open intervals are homeomorphic and any two open intervals are homeomorphic, but [0; 1], [0; 1), and (0; 1) are pairwise non-homeomorphic. Indeed, suppose there is a homeomorphism h : [0; 1] ! (0; 1). Removing the endpoint 0 from [0; 1] leaves the closed interval connected, but removing h(0) from (0; 1) neces- sarily disconnects the open interval. Similarly the other two pairs are non- homeomorphic. The half-open interval can be continuously and bijectively mapped to the circle S1 = fx 2 R2 j kxk = 1g. Indeed, f : [0; 1) ! S1 defined by f(x) = (sin 2πx; cos 2πx) and illustrated in Figure I.7 is such a map. Removing the midpoint disconnects any one of the three intervals but f 0 1 Figure I.7: A continuous bijection whose inverse is not continuous.
    [Show full text]
  • Knot Genus Recall That Oriented Surfaces Are Classified by the Either Euler Characteristic and the Number of Boundary Components (Assume the Surface Is Connected)
    Knots and how to detect knottiness. Figure-8 knot Trefoil knot The “unknot” (a mathematician’s “joke”) There are lots and lots of knots … Peter Guthrie Tait Tait’s dates: 1831-1901 Lord Kelvin (William Thomson) ? Knots don’t explain the periodic table but…. They do appear to be important in nature. Here is some knotted DNA Science 229, 171 (1985); copyright AAAS A 16-crossing knot one of 1,388,705 The knot with archive number 16n-63441 Image generated at http: //knotilus.math.uwo.ca/ A 23-crossing knot one of more than 100 billion The knot with archive number 23x-1-25182457376 Image generated at http: //knotilus.math.uwo.ca/ Spot the knot Video: Robert Scharein knotplot.com Some other hard unknots Measuring Topological Complexity ● We need certificates of topological complexity. ● Things we can compute from a particular instance of (in this case) a knot, but that does change under deformations. ● You already should know one example of this. ○ The linking number from E&M. What kind of tools do we need? 1. Methods for encoding knots (and links) as well as rules for understanding when two different codings are give equivalent knots. 2. Methods for measuring topological complexity. Things we can compute from a particular encoding of the knot but that don’t agree for different encodings of the same knot. Knot Projections A typical way of encoding a knot is via a projection. We imagine the knot K sitting in 3-space with coordinates (x,y,z) and project to the xy-plane. We remember the image of the projection together with the over and under crossing information as in some of the pictures we just saw.
    [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]