DOCSLIB.ORG

JMM 2016 Student Poster Session Abstract Book

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Abstracts for the MAA Undergraduate Poster Session Seattle, WA January 8, 2016 Organized by Joyati Debnath Winona State University CELEBRATING A CENTURY OF ADVANCING MATHEMATICS Organized by the MAA Committee on Undergraduate Student Activities and Chapters and CUPM Subcommittee on Research by Undergraduates Dear Students, Advisors, Judges and Colleagues, If you look around today you will see about 341 posters and 551 presenters, record numbers, once again. It is so rewarding to see this session, which offers such a great opportunity for interaction between students and professional mathematicians, continue to grow. The judges you see here today are professional mathematicians from institutionsaround the world. They are advisors, colleagues, new Ph.D.s, and administrators. We have acknowledged many of them in this booklet; however, many judges here volunteered on site. Their support is vital to the success of the session and we thank them. We are supported financially by the National Science Foundation, Tudor Investments, and Two Sigma. We are also helped by the members of the Committee on Undergraduate Student Activities and Chapters (CUSAC) in some way or other. They are: Jiehua Zhu; Pamela A. Richardson; Jennifer Schaefer; Lisa Marano; Dora C. Ahmadi; Andy Niedermaier; Benjamin Galluzzo; Eve Torrence; Gerard A. Venema; Jen- nifer Bergner; Jim Walsh; Kristina Cole Garrett; May Mei; Dr. Richard Neal; TJ Hitchman; William J. Higgins; and Zsuzsanna Szaniszlo. There are many details of the poster session that begin with putting out the advertisement in FOCUS, ensuring students have travel money, online submission work properly, and organizing poster boards and tables in the room we are in today that are attributed to Gerard Venema (MAA Associate Secretary), Linda Braddy (MAA), and Penny Pina (AMS). Our online submission system and technical support is key to managing the ever-growing number of poster entries we receive. Thanks to MAA staff, especially Margaret Maurer and Maia Henley, for their work setting up and managing the system this year. Preparation of the abstract book is a time-consuming task. Thanks to Beverly Ruedi for doing the final production work on the abstract book. Thank yous go to Angel R. Pineda (California State University, Fullerton), James P. Solazzo (Coastal Carolina University), Rebecca Garcia (Sam Houston State University), and Dora Ahmadi (Moorehead State University) for organizing an orientation for the judges and authoring the judging form. James also helped reviewing several abstracts. Thanks to all the students, judges, volunteers, and sponsors. I hope you have a wonderful experience at this years poster session! Joyati Debnath Winona State University 1 The MAA gratefully acknowledges the support of Tudor Investment Corporation and Two Sigma. Their generosity on behalf of the 2016 Undergraduate Student Poster Session enables students to interact with peers and role models in the mathematical sciences during the largest mathematics meeting in the world. Support for the student travel grant is provided by the National Science Foundation (grant DMS-1341843). Judges and Judge Affiliations 1. Caleb Adams, Radford University 2. Stephen Aldrich, Adams State University 3. Waleed Al-Rawashdeh, Montana Tech 4. Jacqueline Anderson, Bridgewater State University 5. Rene Ardila, University of Iowa 6. Jathan Austin, Salisbury University 7. Chad Awtrey, Elon University 8. Mohammad K. Azarian, University of Evansville 9. Liljana Babinkostova, Boise State University 10. Eric Bahuaud, Seattle University 11. Jeremiah Bartz, Francis Marion University 12. Jenn Berg, Fitchburg State University 13. Julie Bergner, University of California, Riverside 14. Joanna Bieri, Universtiy of Redlands 15. Karen Bliss, Virginia Military Institute 16. Rachelle Bouchat, Indiana University of Pennsylvania 17. David Brown, Ithaca College 18. Marko Budisic, University of Wisconsin, Madison 19. Neal Bushaw, Arizona State University 20. Frederick Butler, York College of Pennsylvania 21. Alex Capaldi, Valparaiso University 22. Carmen Caprau, California State University, Fresno 23. Rodica Cazacu, Georgia College 24. Vinodh Kumar Chellamuthu, Dixie State University 25. Po-Keng Cheng, Stony Brook University/Applied Mathematics and Statistics 26. David Clark, Grand Valley State University 27. Benjamin Collins, University of Wisconsin-Platteville 28. Catherine Crockett, Point Loma Nazarene University 29. Cheryll Crowe, Asbury University 30. Aldo Cruz-Cota, Texas Wesleyan University 31. Ulrich Daepp, Bucknell University 32. Dan Daly, Southeast Missouri State University 33. Guy David, New York University 34. Rachel Davis, Purdue University 35. Jessica De Silva, University of Nebraska-Lincoln 36. Mihiri De Silva 37. Meghan De Witt, St Thomas Aquinas College 38. John Diamantopoulos, Northeastern State University 39. Elizabeth Donovan, Murray State University 40. Cecilia Dorado, East Tennessee State University 41. Suzanne Doree, Augsburg College 42. Colleen Duffy, University of Wisconsin - Eau Claire 3 4 Judges and Judge Affiliations 43. Julia Eaton, University of Washington Tacoma 44. Tom Edgar, Pacific Lutheran University 45. Joe Eichholz, Rose-Hulman Institute of Technology 46. Henry Escuadro, Juniata College 47. Eleanor Farrington, Massachusetts Maritime Academy 48. Joshua Fetbrandt, Western Nevada College 49. Mary Flagg, University of St. Thomas, Houston 50. Tim Flowers, Indiana University of Pennsylvania 51. Norman Fox, Austin Peay State University 52. Amanda Francis, Brigham Young University 53. Brendan Fry, University of Colorado Boulder 54. Ryan Gantner, St. John Fisher College 55. John Gemmer, Brown University 56. Adrian Gentle, University of Southern Indiana 57. Whitney George, University of Wisconsin-La Crosse 58. Petre Ghenciu, University of Wisconsin-Stout 59. Sayonita Ghosh Hajra, University of Utah 60. Eva Goedhart, Smith College 61. Heidi Goodson, University of Minnesota 62. William Green, Rose-Hulman Institute of Technology 63. Raymond N. Greenwell, Hofstra University 64. Neha Gupta, Univ. of Illinois Urbana-Champaign 65. Spencer Hamblen, McDaniel College 66. Carl Hammarsten, Lafayette College 67. Marshall Hampton, University of Minnesota Duluth 68. Qi Han, Worcester Polytechnic Institute 69. Amanda Harsy, Lewis University 70. James Hartman, The College of Wooster 71. Katie Haymaker, Villanova University 72. Alan Haynes, University of York (UK) 73. Allison Henrich, Seattle University 74. Edwin (Jed) Herman, University of Wisconsin-Stevens Point 75. Aparna Higgins, University of Dayton 76. Firas Hindeleh, Grand Valley State University 77. Thomas Hoft, University of St. Thomas 78. Joshua Holden, Rose-Hulman Institute of Technology 79. Werner Horn, California State University, Northridge 80. Tingting Huan, College of the Holy Cross 81. Leonard Huang, University of Kansas 82. Kevin Iga, Pepperdine University 83. Garth Isaak, Lehigh University 84. Michael Jackson, Grove City College 85. Jin Woo Jang, University of Pennsylvania 86. Janine Janoski, King’s College 87. Mike Janssen, Dordt College Judges and Judge Affiliations 5 88. Rasitha Jayasekare, Butler University 89. Jonathan Jedwab, Simon Fraser University 90. Mohammed Kaabar, Washington State University 91. Judit Kardos, TCNJ 92. Sanjeewa Karunarathna, Texas Tech University 93. Mitch Keller, Washington and Lee University 94. Michael Kelly, The Ohio State University 95. James Kelly, Christopher Newport University 96. Franklin Kenter, Rice University 97. Lauren Keough, Davidson College 98. Noureen Khan, University of North Texas at Dallas 99. Vicky Klima, Appalachian State University 100. Alan Koch, Agnes Scott College 101. Vlajko Kocic, Xavier University of Louisiana 102. Lucas Kramer, Bethel College 103. Wei-Kai Lai, University of South Carolina Salkehatchie 104. Liz Lane-Harvard, University of Central Oklahoma 105. Kristin Lassonde, Klamath Community College 106. Sheldon Lee, Sheldon Lee 107. Mary Margarita Legner, Riverside City College 108. Andrzej Lenard 109. Jessie Lenarz, St. Catherine University 110. Shannon Lockard, Bridgewater State University 111. Yun Lu, Kutztown University 112. Timothy Lucas, Pepperdine University 113. Jason Lutz, University of Nebraska-Lincoln 114. Teresa Magnus, Rivier University 115. Sara Malec, Hood College 116. Marco V Martinez, North Central College 117. Maeve McCarthy, Murray State University 118. Maarten McKubre-Jordens, University of Canterbury 119. Patricia Mellodge, University of Hartford 120. Kristi Meyer, Wisconsin Lutheran College 121. David Milan, UT Tyler 122. Charles Moore, Washington State University 123. Benjamin Morin, Arizona State University 124. Jeremy Muskat, Western State Colorado University 125. Kellen Myers, Farmingdale State College, SUNY 126. Dawn Nelson, Saint Peter’s University 127. Jeffrey Neugebauer, Eastern Kentucky University 128. Maria Nogin, California State University, Fresno 129. David Offner, Westminster College 130. Seungly Oh, Western New England University 131. Carl Olimb, Augustana University 132. Paul Olson, Penn State Erie , Behrend College 6 Judges and Judge Affiliations 133. Peter Olszewski, Penn State Behrend 134. Christopher ONeill, Texas A&M University 135. Frank Patane, Samford University 136. Katherine Paullin,University of Kentucky 137. Pei Pei, Earlham College 138. James Peirce, University of Wisconsin - La Crosse 139. Timothy Pennings, Davenport University 140. Leonardo Pinheiro, Rhode Island College 141. Geremias Polanco Encarnacion, Hampshire College 142. Megan Powell, University of St. Francis 143.
Recommended publications
  • Hyperbolic 4-Manifolds and the 24-Cell

    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.
  • Limiting Values and Functional and Difference Equations †

    Limiting Values and Functional and Difference Equations †

    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan; [email protected] * Correspondence: [email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions.
  • Dehn Surgery on Knots in S^ 3 Producing Nil Seifert Fibred Spaces

    Dehn Surgery on Knots in S^ 3 Producing Nil Seifert Fibred Spaces

    Dehn surgery on knots in S3 producing Nil Seifert fibred spaces Yi Ni Department of Mathematics, Caltech 1200 E California Blvd, Pasadena, CA 91125 Email: [email protected] Xingru Zhang Department of Mathematics, University at Buffalo Email: xinzhang@buffalo.edu Abstract We prove that there are exactly 6 Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in S3, with {60, 144, 156, 288, 300} as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly 4 specific hyperbolic knots in S3 which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in S3. 1 Introduction 3 3 For a knot K in S , we denote by SK (p/q) the manifold obtained by Dehn surgery along K with slope p/q. Here the slope p/q is parameterized by the standard meridian/longitude coordinates of K and we always assume gcd(p, q) = 1. In this paper we study the problem of on which knots in S3 with which slopes Dehn surgeries can produce Seifert fibred spaces admitting the Nil geometry. Recall that every closed connected orientable Seifert fibred space W admits one of 6 canonical geometries: S2 R, E3, 2 3 × H R, S , Nil, SL (R). More concretely if e(W ) denotes the Euler number of W and χ( W ) denotes × 2 B the orbifold Euler characteristic of the base orbifold W of W , then the geometry of W is uniquely B determined by thef values of e(W ) and χ( W ) according to the following table (cf.
  • Deep Learning the Hyperbolic Volume of a Knot Arxiv:1902.05547V3 [Hep-Th] 16 Sep 2019

    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.
  • An Analysis of Primality Testing and Its Use in Cryptographic Applications

    An Analysis of Primality Testing and Its Use in Cryptographic Applications

    An Analysis of Primality Testing and Its Use in Cryptographic Applications Jake Massimo Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Information Security Royal Holloway, University of London 2020 Declaration These doctoral studies were conducted under the supervision of Prof. Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathe- matics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Jake Massimo April, 2020 2 Abstract Due to their fundamental utility within cryptography, prime numbers must be easy to both recognise and generate. For this, we depend upon primality testing. Both used as a tool to validate prime parameters, or as part of the algorithm used to generate random prime numbers, primality tests are found near universally within a cryptographer's tool-kit. In this thesis, we study in depth primality tests and their use in cryptographic applications. We first provide a systematic analysis of the implementation landscape of primality testing within cryptographic libraries and mathematical software. We then demon- strate how these tests perform under adversarial conditions, where the numbers being tested are not generated randomly, but instead by a possibly malicious party. We show that many of the libraries studied provide primality tests that are not pre- pared for testing on adversarial input, and therefore can declare composite numbers as being prime with a high probability.
  • THE JONES SLOPES of a KNOT Contents 1. Introduction 1 1.1. The

    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.
  • FACTORING COMPOSITES TESTING PRIMES Amin Witno

    FACTORING COMPOSITES TESTING PRIMES Amin Witno

    WON Series in Discrete Mathematics and Modern Algebra Volume 3 FACTORING COMPOSITES TESTING PRIMES Amin Witno Preface These notes were used for the lectures in Math 472 (Computational Number Theory) at Philadelphia University, Jordan.1 The module was aborted in 2012, and since then this last edition has been preserved and updated only for minor corrections. Outline notes are more like a revision. No student is expected to fully benefit from these notes unless they have regularly attended the lectures. 1 The RSA Cryptosystem Sensitive messages, when transferred over the internet, need to be encrypted, i.e., changed into a secret code in such a way that only the intended receiver who has the secret key is able to read it. It is common that alphabetical characters are converted to their numerical ASCII equivalents before they are encrypted, hence the coded message will look like integer strings. The RSA algorithm is an encryption-decryption process which is widely employed today. In practice, the encryption key can be made public, and doing so will not risk the security of the system. This feature is a characteristic of the so-called public-key cryptosystem. Ali selects two distinct primes p and q which are very large, over a hundred digits each. He computes n = pq, ϕ = (p − 1)(q − 1), and determines a rather small number e which will serve as the encryption key, making sure that e has no common factor with ϕ. He then chooses another integer d < n satisfying de % ϕ = 1; This d is his decryption key. When all is ready, Ali gives to Beth the pair (n; e) and keeps the rest secret.
  • Primality Test

    Primality Test

    Primality test A primality test is an algorithm for determining whether (6k + i) for some integer k and for i = −1, 0, 1, 2, 3, or 4; an input number is prime. Amongst other fields of 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k mathematics, it is used for cryptography. Unlike integer + 3). So a more efficient method is to test if n is divisible factorization, primality tests do not generally give prime by 2 or 3, then to check through all the numbers of form p factors, only stating whether the input number is prime 6k ± 1 ≤ n . This is 3 times as fast as testing all m. or not. Factorization is thought to be a computationally Generalising further, it can be seen that all primes are of difficult problem, whereas primality testing is compara- the form c#k + i for i < c# where i represents the numbers tively easy (its running time is polynomial in the size of that are coprime to c# and where c and k are integers. the input). Some primality tests prove that a number is For example, let c = 6. Then c# = 2 · 3 · 5 = 30. All prime, while others like Miller–Rabin prove that a num- integers are of the form 30k + i for i = 0, 1, 2,...,29 and k ber is composite. Therefore the latter might be called an integer. However, 2 divides 0, 2, 4,...,28 and 3 divides compositeness tests instead of primality tests. 0, 3, 6,...,27 and 5 divides 0, 5, 10,...,25.
  • L-Space Surgery and Twisting Operation

    L-Space Surgery and Twisting Operation

    Algebraic & Geometric Topology 16 (2016) 1727–1772 msp L-space surgery and twisting operation KIMIHIKO MOTEGI A knot in the 3–sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, ie a rational homology 3–sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family Kn . We give a sufficient condition for Kn f g f g to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family Kn contains infinitely many hyperbolic L-space knots. We also demonstrate that f g there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one. 57M25, 57M27; 57N10 1 Introduction Heegaard Floer theory (with Z=2Z coefficients) associates a group HFc.M; t/ to a c c closed, orientable spin 3–manifold .M; t/. The direct sum of HFc.M; t/ for all spin structures is denoted by HFc.M /. A rational homology 3–sphere M is called an c c L-space if HFc.M; t/ is isomorphic to Z=2Z for all spin structures t Spin .M /. 2 Equivalently, the dimension dimZ=2Z HFc.M / is equal to the order H1.M Z/ .A j I j knot K in the 3–sphere S 3 is called an L-space knot if the result K.r/ of r –surgery on K is an L-space for some nonzero integer r , and the pair .K; r/ is called an L-space surgery.
  • On the Number of Unknot Diagrams Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar

    On the Number of Unknot Diagrams Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar

    On the number of unknot diagrams Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar To cite this version: Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar. On the number of unknot diagrams. SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2019, 33 (1), pp.306-326. 10.1137/17M115462X. hal-02049077 HAL Id: hal-02049077 https://hal.archives-ouvertes.fr/hal-02049077 Submitted on 26 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the number of unknot diagrams Carolina Medina1, Jorge L. Ramírez-Alfonsín2,3, and Gelasio Salazar1,3 1Instituto de Física, UASLP. San Luis Potosí, Mexico, 78000. 2Institut Montpelliérain Alexander Grothendieck, Université de Montpellier. Place Eugèene Bataillon, 34095 Montpellier, France. 3Unité Mixte Internationale CNRS-CONACYT-UNAM “Laboratoire Solomon Lefschetz”. Cuernavaca, Mexico. October 17, 2017 Abstract Let D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2n diagrams. A folklore argument shows that at least one of these 2n diagrams is unknot, from which it follows that every diagram has finite unknotting number.
  • Jones Polynomials, Volume, and Essential Knot Surfaces

    Jones Polynomials, Volume, and Essential Knot Surfaces

    KNOTS IN POLAND III BANACH CENTER PUBLICATIONS, VOLUME 100 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2013 JONES POLYNOMIALS, VOLUME AND ESSENTIAL KNOT SURFACES: A SURVEY DAVID FUTER Department of Mathematics, Temple University Philadelphia, PA 19122, USA E-mail: [email protected] EFSTRATIA KALFAGIANNI Department of Mathematics, Michigan State University East Lansing, MI 48824, USA E-mail: [email protected] JESSICA S. PURCELL Department of Mathematics, Brigham Young University Provo, UT 84602, USA E-mail: [email protected] Abstract. This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [18, 19], while this survey focuses on the main ideas and examples. Introduction. To every knot in S3 there corresponds a 3-manifold, namely the knot complement. This 3-manifold decomposes along tori into geometric pieces, where the most typical scenario is that all of S3 r K supports a complete hyperbolic metric [43]. Incompressible surfaces embedded in S3 r K play a crucial role in understanding its classical geometric and topological invariants. The quantum knot invariants, including the Jones polynomial and its relatives, the colored Jones polynomials, have their roots in representation theory and physics [28, 46], 2010 Mathematics Subject Classification:57M25,57M50,57N10. D.F. is supported in part by NSF grant DMS–1007221. E.K. is supported in part by NSF grants DMS–0805942 and DMS–1105843. J.P. is supported in part by NSF grant DMS–1007437 and a Sloan Research Fellowship. The paper is in final form and no version of it will be published elsewhere.
  • On Computation of HOMFLY-PT Polynomials of 2–Bridge Diagrams

    On Computation of HOMFLY-PT Polynomials of 2–Bridge Diagrams

    . On computation of HOMFLY-PT polynomials of 2{bridge diagrams . .. Masahiko Murakami . Joint work with Fumio Takeshita and Seiichi Tani Nihon University . December 20th, 2010 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 1 / 28 Contents Motivation and Results Preliminaries Computation Conclusion 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 2 / 28 Contents Motivation and Results Preliminaries Computation Conclusion 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 3 / 28 There exist polynomial time algorithms for computing Jones polynomials and HOMFLY-PT polynomials under reasonable restrictions. Computational Complexities of Knot Polynomials Alexander polynomial [Alexander](1928) Generally, polynomial time Jones polynomial [Jones](1985) Generally, #P{hard [Jaeger, Vertigan and Welsh](1993) HOMFLY-PT polynomial [Freyd, Yetter, Hoste, Lickorish, Millett, Ocneanu](1985) [Przytycki, Traczyk](1987) Generally, #P{hard [Jaeger, Vertigan and Welsh](1993) 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 4 / 28 Computational Complexities of Knot Polynomials Alexander polynomial [Alexander](1928) Generally, polynomial time Jones polynomial [Jones](1985) Generally, #P{hard [Jaeger, Vertigan and Welsh](1993) HOMFLY-PT polynomial [Freyd, Yetter, Hoste, Lickorish, Millett, Ocneanu](1985) [Przytycki, Traczyk](1987) Generally, #P{hard [Jaeger, Vertigan