The Conway Knot Is Not Slice
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Slicing Bing Doubles
Algebraic & Geometric Topology 6 (2006) 3001–999 3001 Slicing Bing doubles DAVID CIMASONI Bing doubling is an operation which produces a 2–component boundary link B.K/ from a knot K. If K is slice, then B.K/ is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if B.K/ is boundary slice, then K is algebraically slice. We also show that the Rasmussen invariant can tell that certain Bing doubles are not smoothly slice. 57M25; 57M27 Introduction Bing doubling [2] is a standard construction which, given a knot K in S 3 , produces a 2–component oriented link B.K/ as illustrated below. (See Section 1 for a precise definition.) K B.K/ Figure 1: The figure eight knot and its Bing double It is easy to check that if the knot K is slice, then the link B.K/ is slice. Does the converse hold ? An affirmative answer to this question seems out of reach. However, a result obtained independently by S Harvey [9] and P Teichner [22] provides a first step in this direction. 1 Recall that the Levine–Tristram signature of a knot K is the function K S ޚ 1 W ! defined as follows: for ! S , K .!/ is given by the signature of the Hermitian 2 matrix .1 !/A .1 !/A , where A is a Seifert matrix for K and A denotes the C transposed matrix. Published: December 2006 DOI: 10.2140/agt.2006.6.3001 3002 David Cimasoni Theorem (Harvey [9], Teichner [22]) If B.K/ is slice, then the integral over S 1 of the Levine–Tristram signature of K is zero. -
Conservation of Complex Knotting and Slipknotting Patterns in Proteins
Conservation of complex knotting and PNAS PLUS slipknotting patterns in proteins Joanna I. Sułkowskaa,1,2, Eric J. Rawdonb,1, Kenneth C. Millettc, Jose N. Onuchicd,2, and Andrzej Stasiake aCenter for Theoretical Biological Physics, University of California at San Diego, La Jolla, CA 92037; bDepartment of Mathematics, University of St. Thomas, Saint Paul, MN 55105; cDepartment of Mathematics, University of California, Santa Barbara, CA 93106; dCenter for Theoretical Biological Physics , Rice University, Houston, TX 77005; and eCenter for Integrative Genomics, Faculty of Biology and Medicine, University of Lausanne, 1015 Lausanne, Switzerland Edited by* Michael S. Waterman, University of Southern California, Los Angeles, CA, and approved May 4, 2012 (received for review April 17, 2012) While analyzing all available protein structures for the presence of ers fixed configurations, in this case proteins in their native folded knots and slipknots, we detected a strict conservation of complex structures. Then they may be treated as frozen and thus unable to knotting patterns within and between several protein families undergo any deformation. despite their large sequence divergence. Because protein folding Several papers have described various interesting closure pathways leading to knotted native protein structures are slower procedures to capture the knot type of the native structure of and less efficient than those leading to unknotted proteins with a protein or a subchain of a closed chain (1, 3, 29–33). In general, similar size and sequence, the strict conservation of the knotting the strategy is to ensure that the closure procedure does not affect patterns indicates an important physiological role of knots and slip- the inherent entanglement in the analyzed protein chain or sub- knots in these proteins. -
The Khovanov Homology of Rational Tangles
The Khovanov Homology of Rational Tangles Benjamin Thompson A thesis submitted for the degree of Bachelor of Philosophy with Honours in Pure Mathematics of The Australian National University October, 2016 Dedicated to my family. Even though they’ll never read it. “To feel fulfilled, you must first have a goal that needs fulfilling.” Hidetaka Miyazaki, Edge (280) “Sleep is good. And books are better.” (Tyrion) George R. R. Martin, A Clash of Kings “Let’s love ourselves then we can’t fail to make a better situation.” Lauryn Hill, Everything is Everything iv Declaration Except where otherwise stated, this thesis is my own work prepared under the supervision of Scott Morrison. Benjamin Thompson October, 2016 v vi Acknowledgements What a ride. Above all, I would like to thank my supervisor, Scott Morrison. This thesis would not have been written without your unflagging support, sublime feedback and sage advice. My thesis would have likely consisted only of uninspired exposition had you not provided a plethora of interesting potential topics at the start, and its overall polish would have likely diminished had you not kept me on track right to the end. You went above and beyond what I expected from a supervisor, and as a result I’ve had the busiest, but also best, year of my life so far. I must also extend a huge thanks to Tony Licata for working with me throughout the year too; hopefully we can figure out what’s really going on with the bigradings! So many people to thank, so little time. I thank Joan Licata for agreeing to run a Knot Theory course all those years ago. -
Oriented Pair (S 3,S1); Two Knots Are Regarded As
S-EQUIVALENCE OF KNOTS C. KEARTON Abstract. S-equivalence of classical knots is investigated, as well as its rela- tionship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon’s double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot. 1. Introduction An oriented knot k is a smooth (or PL) oriented pair S3,S1; two knots are regarded as the same if there is an orientation preserving diffeomorphism sending one onto the other. An unoriented knot k is defined in the same way, but without regard to the orientation of S1. Every oriented knot is spanned by an oriented surface, a Seifert surface, and this gives rise to a matrix of linking numbers called a Seifert matrix. Any two Seifert matrices of the same knot are S-equivalent: the definition of S-equivalence is given in, for example, [14, 21, 11]. It is the equivalence relation generated by ambient surgery on a Seifert surface of the knot. In [19], two oriented knots are defined to be S-equivalent if their Seifert matrices are S- equivalent, and the following result is proved. Theorem 1. Two oriented knots are S-equivalent if and only if they are related by a sequence of doubled-delta moves shown in Figure 1. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. .... .... .... .... .... .... .... .... .... ... -
Mutant Knots and Intersection Graphs 1 Introduction
Mutant knots and intersection graphs S. V. CHMUTOV S. K. LANDO We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. Conversely, if we have a weight system depending only on the intersection graphs of chord diagrams, then the composition of such a weight system with the Kontsevich invariant determines a knot invariant that does not distinguish mutant knots. Thus, an equivalence between finite order invariants not distinguishing mutants and weight systems depending on intersections graphs only is established. We discuss relationship between our results and certain Lie algebra weight systems. 57M15; 57M25 1 Introduction Below, we use standard notions of the theory of finite order, or Vassiliev, invariants of knots in 3-space; their definitions can be found, for example, in [6] or [14], and we recall them briefly in Section 2. All knots are assumed to be oriented. Two knots are said to be mutant if they differ by a rotation of a tangle with four endpoints about either a vertical axis, or a horizontal axis, or an axis perpendicular to the paper. If necessary, the orientation inside the tangle may be replaced by the opposite one. Here is a famous example of mutant knots, the Conway (11n34) knot C of genus 3, and Kinoshita–Terasaka (11n42) knot KT of genus 2 (see [1]). C = KT = Note that the change of the orientation of a knot can be achieved by a mutation in the complement to a trivial tangle. -
On Spectral Sequences from Khovanov Homology 11
ON SPECTRAL SEQUENCES FROM KHOVANOV HOMOLOGY ANDREW LOBB RAPHAEL ZENTNER Abstract. There are a number of homological knot invariants, each satis- fying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain com- plex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies. In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsv´ath-Szab´oto the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4; 5)-torus knot, determining these up to an ambiguity in a pair of degrees; to deter- mine the Ozsv´ath-Szab´ospectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsv´ath-Szab´ospectral sequences necessarily lower the delta grading for all pretzel knots. 1. Introduction Recent work in the area of the 3-manifold invariants called knot homologies has il- luminated the relationship between Floer-theoretic knot homologies and `quantum' knot homologies. The relationships observed take the form of spectral sequences starting with a quantum invariant and abutting to a Floer invariant. A primary ex- ample is due to Ozsv´athand Szab´o[15] in which a spectral sequence is constructed from Khovanov homology of a knot (with Z=2 coefficients) to the Heegaard-Floer homology of the 3-manifold obtained as double branched cover over the knot. -
Deep Learning the Hyperbolic Volume of a Knot Arxiv:1902.05547V3 [Hep-Th] 16 Sep 2019
Deep Learning the Hyperbolic Volume of a Knot Vishnu Jejjalaa;b , Arjun Karb , Onkar Parrikarb aMandelstam Institute for Theoretical Physics, School of Physics, NITheP, and CoE-MaSS, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa bDavid Rittenhouse Laboratory, University of Pennsylvania, 209 S 33rd Street, Philadelphia, PA 19104, USA E-mail: [email protected], [email protected], [email protected] Abstract: An important conjecture in knot theory relates the large-N, double scaling limit of the colored Jones polynomial JK;N (q) of a knot K to the hyperbolic volume of the knot complement, Vol(K). A less studied question is whether Vol(K) can be recovered directly from the original Jones polynomial (N = 2). In this report we use a deep neural network to approximate Vol(K) from the Jones polynomial. Our network is robust and correctly predicts the volume with 97:6% accuracy when training on 10% of the data. This points to the existence of a more direct connection between the hyperbolic volume and the Jones polynomial. arXiv:1902.05547v3 [hep-th] 16 Sep 2019 Contents 1 Introduction1 2 Setup and Result3 3 Discussion7 A Overview of knot invariants9 B Neural networks 10 B.1 Details of the network 12 C Other experiments 14 1 Introduction Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of these patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. In this work, we apply this idea to invariants in knot theory. -
Ribbon Concordances and Doubly Slice Knots
Ribbon concordances and doubly slice knots Ruppik, Benjamin Matthias Advisors: Arunima Ray & Peter Teichner The knot concordance group Glossary Definition 1. A knot K ⊂ S3 is called (topologically/smoothly) slice if it arises as the equatorial slice of a (locally flat/smooth) embedding of a 2-sphere in S4. – Abelian invariants: Algebraic invari- Proposition. Under the equivalence relation “K ∼ J iff K#(−J) is slice”, the commutative monoid ants extracted from the Alexander module Z 3 −1 A (K) = H1( \ K; [t, t ]) (i.e. from the n isotopy classes of o connected sum S Z K := 1 3 , # := oriented knots S ,→S of knots homology of the universal abelian cover of the becomes a group C := K/ ∼, the knot concordance group. knot complement). The neutral element is given by the (equivalence class) of the unknot, and the inverse of [K] is [rK], i.e. the – Blanchfield linking form: Sesquilin- mirror image of K with opposite orientation. ear form on the Alexander module ( ) sm top Bl: AZ(K) × AZ(K) → Q Z , the definition Remark. We should be careful and distinguish the smooth C and topologically locally flat C version, Z[Z] because there is a huge difference between those: Ker(Csm → Ctop) is infinitely generated! uses that AZ(K) is a Z[Z]-torsion module. – Levine-Tristram-signature: For ω ∈ S1, Some classical structure results: take the signature of the Hermitian matrix sm top ∼ ∞ ∞ ∞ T – There are surjective homomorphisms C → C → AC, where AC = Z ⊕ Z2 ⊕ Z4 is Levine’s algebraic (1 − ω)V + (1 − ω)V , where V is a Seifert concordance group. -
THE JONES SLOPES of a KNOT Contents 1. Introduction 1 1.1. The
THE JONES SLOPES OF A KNOT STAVROS GAROUFALIDIS Abstract. The paper introduces the Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2, 3,n) pretzel knots. Contents 1. Introduction 1 1.1. The degree of the Jones polynomial and incompressible surfaces 1 1.2. The degree of the colored Jones function is a quadratic quasi-polynomial 3 1.3. q-holonomic functions and quadratic quasi-polynomials 3 1.4. The Jones slopes and the Jones period of a knot 4 1.5. The symmetrized Jones slopes and the signature of a knot 5 1.6. Plan of the proof 7 2. Future directions 7 3. The Jones slopes and the Jones period of an alternating knot 8 4. Computing the Jones slopes and the Jones period of a knot 10 4.1. Some lemmas on quasi-polynomials 10 4.2. Computing the colored Jones function of a knot 11 4.3. Guessing the colored Jones function of a knot 11 4.4. A summary of non-alternating knots 12 4.5. The 8-crossing non-alternating knots 13 4.6. -
A Knot-Vice's Guide to Untangling Knot Theory, Undergraduate
A Knot-vice’s Guide to Untangling Knot Theory Rebecca Hardenbrook Department of Mathematics University of Utah Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 1 / 26 What is Not a Knot? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 2 / 26 What is a Knot? 2 A knot is an embedding of the circle in the Euclidean plane (R ). 3 Also defined as a closed, non-self-intersecting curve in R . 2 Represented by knot projections in R . Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 3 / 26 Why Knots? Late nineteenth century chemists and physicists believed that a substance known as aether existed throughout all of space. Could knots represent the elements? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 4 / 26 Why Knots? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 5 / 26 Why Knots? Unfortunately, no. Nevertheless, mathematicians continued to study knots! Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 6 / 26 Current Applications Natural knotting in DNA molecules (1980s). Credit: K. Kimura et al. (1999) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 7 / 26 Current Applications Chemical synthesis of knotted molecules – Dietrich-Buchecker and Sauvage (1988). Credit: J. Guo et al. (2010) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 8 / 26 Current Applications Use of lattice models, e.g. the Ising model (1925), and planar projection of knots to find a knot invariant via statistical mechanics. Credit: D. Chicherin, V.P. -
Khovanov Homology, Its Definitions and Ramifications
KHOVANOV HOMOLOGY, ITS DEFINITIONS AND RAMIFICATIONS OLEG VIRO Uppsala University, Uppsala, Sweden POMI, St. Petersburg, Russia Abstract. Mikhail Khovanov defined, for a diagram of an oriented clas- sical link, a collection of groups labelled by pairs of integers. These groups were constructed as homology groups of certain chain complexes. The Euler characteristics of these complexes are coefficients of the Jones polynomial of the link. The original construction is overloaded with algebraic details. Most of specialists use adaptations of it stripped off the details. The goal of this paper is to overview these adaptations and show how to switch between them. We also discuss a version of Khovanov homology for framed links and suggest a new grading for it. 1. Introduction For a diagram D of an oriented link L, Mikhail Khovanov [7] constructed a collection of groups Hi;j(D) such that X j i i;j K(L)(q) = q (−1) dimQ(H (D) ⊗ Q); i;j where K(L) is a version of the Jones polynomial of L. These groups are constructed as homology groups of certain chain complexes. The Khovanov homology has proved to be a powerful and useful tool in low dimensional topology. Recently Jacob Rasmussen [16] has applied it to problems of estimating the slice genus of knots and links. He defied a knot invariant which gives a low bound for the slice genus and which, for knots with only positive crossings, allowed him to find the exact value of the slice genus. Using this technique, he has found a new proof of Milnor conjecture on the slice genus of toric knots. -
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.