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Hyperbolic 4-Manifolds and the 24-Cell
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Florence Research Universita` degli Studi di Firenze Dipartimento di Matematica \Ulisse Dini" Dottorato di Ricerca in Matematica Hyperbolic 4-manifolds and the 24-cell Leone Slavich Tutor Coordinatore del Dottorato Prof. Graziano Gentili Prof. Alberto Gandolfi Relatore Prof. Bruno Martelli Ciclo XXVI, Settore Scientifico Disciplinare MAT/03 Contents Index 1 1 Introduction 2 2 Basics of hyperbolic geometry 5 2.1 Hyperbolic manifolds . .7 2.1.1 Horospheres and cusps . .8 2.1.2 Manifolds with boundary . .9 2.2 Link complements and Kirby calculus . 10 2.2.1 Basics of Kirby calculus . 12 3 Hyperbolic 3-manifolds from triangulations 15 3.1 Hyperbolic relative handlebodies . 18 3.2 Presentations as a surgery along partially framed links . 19 4 The ambient 4-manifold 23 4.1 The 24-cell . 23 4.2 Mirroring the 24-cell . 24 4.3 Face pairings of the boundary 3-strata . 26 5 The boundary manifold as link complement 33 6 Simple hyperbolic 4-manifolds 39 6.1 A minimal volume hyperbolic 4-manifold with two cusps . 39 6.1.1 Gluing of the boundary components . 43 6.2 A one-cusped hyperbolic 4-manifold . 45 6.2.1 Glueing of the boundary components . 46 1 Chapter 1 Introduction In the early 1980's, a major breakthrough in the field of low-dimensional topology was William P. Thurston's geometrization theorem for Haken man- ifolds [16] and the subsequent formulation of the geometrization conjecture, proven by Grigori Perelman in 2005. -
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. -
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. -
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. -
Arxiv:1501.00726V2 [Math.GT] 19 Sep 2016 3 2 for Each N, Ln Is Assembled from a Tangle S in B , N Copies of a Tangle T in S × I, and the Mirror Image S of S
HIDDEN SYMMETRIES VIA HIDDEN EXTENSIONS ERIC CHESEBRO AND JASON DEBLOIS Abstract. This paper introduces a new approach to finding knots and links with hidden symmetries using \hidden extensions", a class of hidden symme- tries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of some hyperbolic link complements. A hidden symmetry of a manifold M is a homeomorphism of finite-degree covers of M that does not descend to an automorphism of M. By deep work of Mar- gulis, hidden symmetries characterize the arithmetic manifolds among all locally symmetric ones: a locally symmetric manifold is arithmetic if and only if it has infinitely many \non-equivalent" hidden symmetries (see [13, Ch. 6]; cf. [9]). Among hyperbolic knot complements in S3 only that of the figure-eight is arith- metic [10], and the only other knot complements known to possess hidden sym- metries are the two \dodecahedral knots" constructed by Aitchison{Rubinstein [1]. Whether there exist others has been an open question for over two decades [9, Question 1]. Its answer has important consequences for commensurability classes of knot complements, see [11] and [2]. The partial answers that we know are all negative. Aside from the figure-eight, there are no knots with hidden symmetries with at most fifteen crossings [6] and no two-bridge knots with hidden symmetries [11]. Macasieb{Mattman showed that no hyperbolic (−2; 3; n) pretzel knot, n 2 Z, has hidden symmetries [8]. Hoffman showed the dodecahedral knots are commensurable with no others [7]. -
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. -
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring Of
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring of Pretzel Knots A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Robert Ostrander December 2013 The thesis of Robert Ostrander is approved: |||||||||||||||||| |||||||| Dr. Alberto Candel Date |||||||||||||||||| |||||||| Dr. Terry Fuller Date |||||||||||||||||| |||||||| Dr. Magnhild Lien, Chair Date California State University, Northridge ii Dedications I dedicate this thesis to my family and friends for all the help and support they have given me. iii Acknowledgments iv Table of Contents Signature Page ii Dedications iii Acknowledgements iv Abstract vi Introduction 1 1 Definitions and Background 2 1.1 Knots . .2 1.1.1 Composition of knots . .4 1.1.2 Links . .5 1.1.3 Torus Knots . .6 1.1.4 Reidemeister Moves . .7 2 Properties of Knots 9 2.0.5 Knot Invariants . .9 3 p-Coloring of Pretzel Knots 19 3.0.6 Pretzel Knots . 19 3.0.7 (p1, p2, p3) Pretzel Knots . 23 3.0.8 Applications of Theorem 6 . 30 3.0.9 (p1, p2, p3, p4) Pretzel Knots . 31 Appendix 49 v Abstract P coloring of Pretzel Knots by Robert Ostrander Master of Science in Mathematics In this thesis we give a brief introduction to knot theory. We define knot invariants and give examples of different types of knot invariants which can be used to distinguish knots. We look at colorability of knots and generalize this to p-colorability. We focus on 3-strand pretzel knots and apply techniques of linear algebra to prove theorems about p-colorability of these knots. -
Knot Tricolorability Danielle Brushaber & Mckenzie Hennen
Knot Tricolorability Danielle Brushaber & McKenzie Hennen Faculty Mentor: Carolyn Otto University of Wisconsin-Eau Claire Problem & Importance Definitions Knot Theory, a field of Topology, can be used to Basic Knot Definitions Linear Algebra Definitions model and understand how enzymes (called topoi- Knot- a curve in R3 that is non-intersecting and closed, similar to a circle but somerases) work in DNA processes to untangle or Matrix of a knot- each intersection’s equation determines a row in a matrix. For example, using WH3 , the equation could be tangled up. Consider a tangled string attached at the ends. 1 repair strands of DNA. In a human cell nucleus, the 2a − b − c = 0 would make one row of a matrix. a Knot invariant b DNA is linear, so the knots can slip off the end, - a characteristic of a knot. Useful for distinguishing when two c Determinant- A property of a square matrix. We used the program Maple to find the determinant, as the matrices are n and it is difficult to recognize what the enzymes do. knots are different. Tricolorable-For a knot to be tricolorable, the three lines involved in an intersec- as large as 26 × 26. In the case of the Trefoil knot, finding the determinant by hand is conducted in the following way: a b c First, look at the number in the top left corner and disregard the adjacent column and row. Repeat this step However, the DNA in mito- tion must either be 3 different colors or all 1 color . Tricolorability is a 2 − − =0 2 −1 −1 b a c with the second number in the first row and once more with the last number in the first row. -
Alexander Polynomial, Finite Type Invariants and Volume of Hyperbolic
ISSN 1472-2739 (on-line) 1472-2747 (printed) 1111 Algebraic & Geometric Topology Volume 4 (2004) 1111–1123 ATG Published: 25 November 2004 Alexander polynomial, finite type invariants and volume of hyperbolic knots Efstratia Kalfagianni Abstract We show that given n > 0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order ≤ n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the comple- ment of a hyperbolic alternating knot is bounded above by a linear function of the coefficients of the Alexander polynomial of the knot. As a corollary to our main result we obtain that, for every m> 0, there exists a sequence of hyperbolic knots with trivial finite type invariants of order ≤ m but ar- bitrarily large volume. We discuss how our results fit within the framework of relations between the finite type invariants and the volume of hyperbolic knots, predicted by Kashaev’s hyperbolic volume conjecture. AMS Classification 57M25; 57M27, 57N16 Keywords Alexander polynomial, finite type invariants, hyperbolic knot, hyperbolic Dehn filling, volume. 1 Introduction k i Let c(K) denote the crossing number and let ∆K(t) := Pi=0 cit denote the Alexander polynomial of a knot K . If K is hyperbolic, let vol(S3 \ K) denote the volume of its complement. The determinant of K is the quantity det(K) := |∆K(−1)|. Thus, in general, we have k det(K) ≤ ||∆K (t)|| := X |ci|. (1) i=0 It is well know that the degree of the Alexander polynomial of an alternating knot equals twice the genus of the knot. -
MUTATION of KNOTS Figure 1 Figure 2
proceedings of the american mathematical society Volume 105. Number 1, January 1989 MUTATION OF KNOTS C. KEARTON (Communicated by Haynes R. Miller) Abstract. In general, mutation does not preserve the Alexander module or the concordance class of a knot. For a discussion of mutation of classical links, and the invariants which it is known to preserve, the reader is referred to [LM, APR, MT]. Suffice it here to say that mutation of knots preserves the polynomials of Alexander, Jones, and Homfly, and also the signature. Mutation of an oriented link k can be described as follows. Take a diagram of k and a tangle T with two outputs and two inputs, as in Figure 1. Figure 1 Figure 2 Rotate the tangle about the east-west axis to obtain Figure 2, or about the north-south axis to obtain Figure 3, or about the axis perpendicular to the paper to obtain Figure 4. Keep or reverse all the orientations of T as dictated by the rest of k . Each of the links so obtained is a mutant of k. The reverse k' of a link k is obtained by reversing the orientation of each component of k. Let us adopt the convention that a knot is a link of one component, and that k + I denotes the connected sum of two knots k and /. Lemma. For any knot k, the knot k + k' is a mutant of k + k. Received by the editors September 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. ©1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page 206 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MUTATION OF KNOTS 207 Figure 3 Figure 4 Proof. -
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.