An Introduction to the Volume Conjecture
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Experimental Evidence for the Volume Conjecture for the Simplest Hyperbolic Non-2-Bridge Knot Stavros Garoufalidis Yueheng Lan
ISSN 1472-2739 (on-line) 1472-2747 (printed) 379 Algebraic & Geometric Topology Volume 5 (2005) 379–403 ATG Published: 22 May 2005 Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot Stavros Garoufalidis Yueheng Lan Abstract Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the knot. We provide an efficient formula for the colored Jones function of the simplest hyperbolic non-2-bridge knot, and using this formula, we provide numerical evidence for the Hyperbolic Volume Conjecture for the simplest hyperbolic non-2-bridge knot. AMS Classification 57N10; 57M25 Keywords Knots, q-difference equations, asymptotics, Jones polynomial, Hyperbolic Volume Conjecture, character varieties, recursion relations, Kauffman bracket, skein module, fusion, SnapPea, m082 1 Introduction 1.1 The Hyperbolic Volume Conjecture The Volume Conjecture connects two very different approaches to knot theory, namely Topological Quantum Field Theory and Riemannian (mostly Hyper- bolic) Geometry. The Volume Conjecture states that for every hyperbolic knot K in S3 2πi log |J (n)(e n )| 1 lim K = vol(K), (1) n→∞ n 2π where ± • JK (n) ∈ Z[q ] is the Jones polynomial of a knot colored with the n- dimensional irreducible representation of sl2 , normalized so that it equals to 1 for the unknot (see [J, Tu]), and c Geometry & Topology Publications 380 Stavros Garoufalidis and Yueheng Lan • vol(K) is the volume of a complete hyperbolic metric in the knot com- plement S3 − K ; see [Th]. -
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. -
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. -
Arxiv:Math-Ph/0305039V2 22 Oct 2003
QUANTUM INVARIANT FOR TORUS LINK AND MODULAR FORMS KAZUHIRO HIKAMI ABSTRACT. We consider an asymptotic expansion of Kashaev’s invariant or of the colored Jones function for the torus link T (2, 2 m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of a modular form of weight 3/2 which is related to the su(2)m−2 character. c 1. INTRODUCTION Recent studies reveal an intimate connection between the quantum knot invariant and “nearly modular forms” especially with half integral weight. In Ref. 9 Lawrence and Zagier studied an as- ymptotic expansion of the Witten–Reshetikhin–Turaev invariant of the Poincar´ehomology sphere, and they showed that the invariant can be regarded as the Eichler integral of a modular form of weight 3/2. In Ref. 19, Zagier further studied a “strange identity” related to the half-derivatives of the Dedekind η-function, and clarified a role of the Eichler integral with half-integral weight. From the viewpoint of the quantum invariant, Zagier’s q-series was originally connected with a gener- ating function of an upper bound of the number of linearly independent Vassiliev invariants [17], and later it was found that Zagier’s q-series with q being the N-th root of unity coincides with Kashaev’s invariant [5, 6], which was shown [14] to coincide with a specific value of the colored Jones function, for the trefoil knot. This correspondence was further investigated for the torus knot, and it was shown [3] that Kashaev’s invariant for the torus knot T (2, 2 m + 1) also has a nearly arXiv:math-ph/0305039v2 22 Oct 2003 modular property; it can be regarded as a limit q being the root of unity of the Eichler integral of the Andrews–Gordon q-series, which is theta series with weight 1/2 spanning m-dimensional space. -
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. -
CROSSING CHANGES and MINIMAL DIAGRAMS the Unknotting
CROSSING CHANGES AND MINIMAL DIAGRAMS ISABEL K. DARCY The unknotting number of a knot is the minimal number of crossing changes needed to convert the knot into the unknot where the minimum is taken over all possible diagrams of the knot. For example, the minimal diagram of the knot 108 requires 3 crossing changes in order to change it to the unknot. Thus, by looking only at the minimal diagram of 108, it is clear that u(108) ≤ 3 (figure 1). Nakanishi [5] and Bleiler [2] proved that u(108) = 2. They found a non-minimal diagram of the knot 108 in which two crossing changes suffice to obtain the unknot (figure 2). Lower bounds on unknotting number can be found by looking at how knot invariants are affected by crossing changes. Nakanishi [5] and Bleiler [2] used signature to prove that u(108)=2. Figure 1. Minimal dia- Figure 2. A non-minimal dia- gram of knot 108 gram of knot 108 Bernhard [1] noticed that one can determine that u(108) = 2 by using a sequence of crossing changes within minimal diagrams and ambient isotopies as shown in figure. A crossing change within the minimal diagram of 108 results in the knot 62. We then use an ambient isotopy to change this non-minimal diagram of 62 to a minimal diagram. The unknot can then be obtained by changing one crossing within the minimal diagram of 62. Bernhard hypothesized that the unknotting number of a knot could be determined by only looking at minimal diagrams by using ambient isotopies between crossing changes. -
Ideal Polyhedral Surfaces in Fuchsian Manifolds
Geometriae Dedicata https://doi.org/10.1007/s10711-019-00480-y ORIGINAL PAPER Ideal polyhedral surfaces in Fuchsian manifolds Roman Prosanov1,2,3 Received: 10 July 2018 / Accepted: 26 August 2019 © The Author(s) 2019 Abstract Let Sg,n be a surface of genus g > 1 with n > 0 punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a given conformal class. Keywords Discrete curvature · Alexandrov theorem · Discrete conformality · Discrete uniformization Mathematics Subject Classification 57M50 · 52B70 · 52B10 · 52C26 1 Introduction 1.1 Theorems of Alexandrov and Rivin Consider a convex polytope P ⊂ R3. Its boundary is homeomorphic to S2 and carries the intrinsic metric induced from the Euclidean metric on R3. What are the properties of this metric? AmetriconS2 is called polyhedral Euclidean if it is locally isometric to the Euclidean metric on R2 except finitely many points, which have neighborhoods isometric to an open subset of a cone (an exceptional point is mapped to the apex of this cone). If the conical angle of every exceptional point is less than 2π, then this metric is called convex. It is clear that the induced metric on the boundary of a convex polytope is convex polyhedral Euclidean. One can ask a natural question: is this description complete, in the sense that every convex polyhedral Euclidean metric can be realized as the induced metric of a polytope? This question was answered positively by Alexandrov in 1942, see [2,3]: Supported by SNF Grant 200021−169391 “Discrete curvature and rigidity”. -
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. -
Volume Conjecture and Topological Recursion
. Volume Conjecture and Topological Recursion .. Hiroyuki FUJI . Nagoya University Collaboration with R.H.Dijkgraaf (ITFA) and M. Manabe (Nagoya Math.) 6th April @ IPMU Papers: R.H.Dijkgraaf and H.F., Fortsch.Phys.57(2009),825-856, arXiv:0903.2084 [hep-th] R.H.Dijkgraaf, H.F. and M.Manabe, to appear. Hiroyuki FUJI Volume Conjecture and Topological Recursion . 1. Introduction Asymptotic analysis of the knot invariants is studied actively in the knot theory. Volume Conjecture [Kashaev][Murakami2] Asymptotic expansion of the colored Jones polynomial for knot K ) The geometric invariants of the knot complement S3nK. S3 Recent years the asymptotic expansion is studied to higher orders. 2~ Sk(u): Perturbative" invariants, q = e [Dimofte-Gukov-Lenells-Zagier]# X1 1 δ k Jn(K; q) = exp S0(u) + log ~ + ~ Sk+1(u) ; u = 2~n − 2πi: ~ 2 k=0 Hiroyuki FUJI Volume Conjecture and Topological Recursion Topological Open String Topological B-model on the local Calabi-Yau X∨ { } X∨ = (z; w; ep; ex) 2 C2 × C∗ × C∗jH(ep; ex) = zw : D-brane partition function ZD(ui) Calab-Yau manifold X Disk World-sheet Labrangian Embedding submanifold = D-brane Ending on Lag.submfd. Holomorphic disk Hiroyuki FUJI Volume Conjecture and Topological Recursion Topological Recursion [Eynard-Orantin] Eynard and Orantin proposed a spectral invariants for the spectral curve C C = f(x; y) 2 (C∗)2jH(y; x) = 0g: • Symplectic structure of the spectral curve C, • Riemann surface Σg,h = World-sheet. (g,h) ) Spectral invariant F (u1; ··· ; uh) ui:open string moduli Eynard-Orantin's topological recursion is applicable. x x1 xh x x x x x x 1 h x x k1 k i1 ij hj q q q = q + Σ l J g g 1 g l l Σg,h+1 Σg−1,h+2 Σℓ,k+1 Σg−ℓ,h−k Spectral invariant = D-brane free energy in top. -
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.