Knot Theory: History and Applications with a Connection to Graph Theory
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Scottish Physics and Knot Theory's Odd Origins
Scottish Physics and Knot Theory’s Odd Origins Daniel S. Silver My mother, a social worker and teacher, encouraged my interest in the mysteries of thought. My father, a physical chemist, fostered my appreciation of the history of science. The former chair of my department, prone to unguarded comment, once accused me of being too young to think about such things. As the years pass, I realize how wrong he is. Introduction. Knot theory is one of the most active research areas of mathematics today. Thousands of refereed articles about knots have been published during just the past ten years. One publication, Journal of Knot Theory and its Ramifications, published by World Scientific since 1992, is devoted to new results in the subject. It is perhaps an oversimplification, but not a grave one, to say that two mildly eccentric nineteenth century Scottish physicists, William Thomson (a.k.a. Lord Kelvin) and Peter Guthrie Tait were responsible for modern knot theory. Our purpose is not to debate the point. Rather it is to try to understand the beliefs and bold expectations that motivated Thomson and Tait. James Clerk Maxwell’s influence on his friend Tait is well known. However, Maxwell’s keen interest in the developing theory of knots and links has become clear only in recent years, thanks in great part to the efforts of M. Epple. Often Maxwell was a source of information and humorous encouragement. At times he attempted to restrain Tait’s flights from the path of science. Any discussion of the origin of knot theory requires consideration of the colorful trefoil of Scottish physicists: Thomsom, Tait and Maxwell. -
Links, Quantum Groups, and TQFT's
Abstract. The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given. The quantum group Uq(sl2), which gives rise to the Jones polynomial, is constructed explicitly. The 3-manifold invariants and the axiomatic topological quantum field theories which arise from these link invariants at certain values of the parameter are constructed. LINKS, QUANTUM GROUPS AND TQFT’S STEPHEN SAWIN Introduction In studying a class of mathematical objects, such as knots, one usually begins by developing ideas and machinery to understand specific features of them. When invariants are discovered, they arise out of such understanding and generally give information about those features. Thus they are often immediately useful for proving theorems, even if they are difficult to compute. This is not the situation we find ourselves in with the invariants of knots and 3- manifolds which have appeared since the Jones polynomial. We have a wealth of invariants, all readily computable, but standing decidedly outside the traditions of knot and 3-manifold theory. They lack a geometric interpretation, and consequently have been of almost no use in answering questions one might have asked before their creation. In effect, we are left looking for the branch of mathematics from which these should have come organically. In fact, we have an answer of sorts. In [Wit89b], Witten gave a heuristic definition of the Jones polynomial in terms of a topological quantum field theory, following the arXiv:q-alg/9506002v2 31 Jul 1995 outline of a program proposed by Atiyah [Ati88]. -
KNOTS from Combinatorics of Knot Diagrams to Combinatorial Topology Based on Knots
2034-2 Advanced School and Conference on Knot Theory and its Applications to Physics and Biology 11 - 29 May 2009 KNOTS From combinatorics of knot diagrams to combinatorial topology based on knots Jozef H. Przytycki The George Washington University Washington DC USA KNOTS From combinatorics of knot diagrams to combinatorial topology based on knots Warszawa, November 30, 1984 { Bethesda, March 3, 2007 J´ozef H. Przytycki LIST OF CHAPTERS: Chapter I: Preliminaries Chapter II: History of Knot Theory This e-print. Chapter II starts at page 3 Chapter III: Conway type invariants Chapter IV: Goeritz and Seifert matrices Chapter V: Graphs and links e-print: http://arxiv.org/pdf/math.GT/0601227 Chapter VI: Fox n-colorings, Rational moves, Lagrangian tangles and Burnside groups Chapter VII: Symmetries of links Chapter VIII: Different links with the same Jones type polynomials Chapter IX: Skein modules e-print: http://arxiv.org/pdf/math.GT/0602264 2 Chapter X: Khovanov Homology: categori- fication of the Kauffman bracket relation e-print: http://arxiv.org/pdf/math.GT/0512630 Appendix I. Appendix II. Appendix III. Introduction This book is about classical Knot Theory, that is, about the position of a circle (a knot) or of a number of disjoint circles (a link) in the space R3 or in the sphere S3. We also venture into Knot Theory in general 3-dimensional manifolds. The book has its predecessor in Lecture Notes on Knot Theory, which were published in Polish1 in 1995 [P-18]. A rough translation of the Notes (by J.Wi´sniewski) was ready by the summer of 1995. -
A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas
A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas Compiled by Thomas Zaslavsky Department of Mathematical Sciences Binghamton University (SUNY) Binghamton, New York, U.S.A. 13902-6000 E-mail: [email protected] Submitted: March 19, 1998; Accepted: July 20, 1998. Eighth Edition 6 September 2012 Mathematics Subject Classifications (2000): Primary 05-00, 05-02, 05C22; Secondary 05B20, 05B35, 05C07, 05C10, 05C15, 05C17, 05C20, 05C25, 05C30, 05C35, 05C38, 05C40, 05C45, 05C50, 05C60, 05C62, 05C65, 05C70, 05C75, 05C80, 05C83, 05C85, 05C90, 05C99, 05E25, 05E30, 06A07, 15A06, 15A15, 15A39, 15A99, 20B25, 20F55, 34C99, 51D20, 51D35, 51E20, 51M09, 52B12, 52C07, 52C35, 57M27, 68Q15, 68Q25, 68R10, 82B20, 82D30, 90B10, 90C08, 90C27, 90C35, 90C57, 90C60, 91B14, 91C20, 91D30, 91E10, 92D40, 92E10, 94B75. Colleagues: HELP! If you have any suggestions whatever for items to include in this bibliography, or for other changes, please let me hear from you. Thank you. Copyright c 1996, 1998, 1999, 2003{2005, 2009{2012 Thomas Zaslavsky the electronic journal of combinatorics #DS8 ii Index A 1 H 135 O 227 V 301 B 22 I 160 P 230 W 306 C 54 J 163 Q 241 X 317 D 77 K 171 R 242 Y 317 E 91 L 188 S 255 Z 321 F 101 M 201 T 288 G 113 N 221 U 301 [I]t should be borne in mind that incompleteness is a necessary concomitant of every collection of whatever kind. Much less can completeness be expected in a first collection, made by a single individual, in his leisure hours, and in a field which is already boundless and is yet expanding day by day. -
Classical Roots of Knot Theory
Chnos, Solrnons & Fracrals, Vol. 9. No. 415, pp. 531-545. 1YY8 Pergamon 0 1998 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-0779198 $19.02 --0.00 PII: SO960-0779(97)00107-O Classical Roots of Knot Theory J6ZEF H. PRZYTYCKIT Department of Mathematics, George Washington University, Washington, DC 20052, USA Abstract-Vandermonde wrote in 1771: “Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced.” We sketch in this essay the history of knot theory stressing the combinatorial aspect of the theory that is so visible in Jones type invariants. fQ 1998 Elsevier Science Ltd. All rights reserved Next he [Alexander] marched into Pisidia where he subdued any resistance which he encountered, and then made himself master of Phrygia. When he captured Gordium [in March 333 BC] which is reputed to have been the home of the ancient king Midas, he saw the celebrated chariot which was fastened to its yoke by the bark of the cornel-tree, and heard the legend which was believed by all barbarians that the fates had decreed that the man who untied the knot was destined to become the ruler of the whole world. According to most writers, the fastenings were so elaborately intertwined and coiled upon one another that their ends were hidden: in consequence Alexander did not know what to do, and in the end loosened the knot by cutting through it with his sword, whereupon the many ends sprang into view. -
Graph Theory Graph Theory (III)
J.A. Bondy U.S.R. Murty Graph Theory (III) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. -
Chapter 1. a Century of Knot Theory 1 Chapter 1
Chapter 1. A Century of Knot Theory 1 Chapter 1. A Century of Knot Theory Note. In this section we give some history of knot theory. Some of this information is from the Wikipedia page on ”Knot Theory” and the Wikipedia page on “The History of Knot Theory” (accessed 1/27/2021). Note. A mathematical theory of knots was first discussed by Alexandre-Th´eophile Vandermonde (February 28, 1735–January 1, 1796) in 1771 in his Remarques sur des Probl`emesde Situation. In this, he remarked that the topological properties are the important properties in discussing knots (interpret “situation” as “position”): “Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with ques- tions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced.” (from Wikipedia page on Vandermonde) Note. The linking number of two knots is related to the way the knots are inter- twined: From the Wikipedia “Linking Number” Page. Chapter 1. A Century of Knot Theory 2 In 1833, Johann Carl Friedrich Gauss (April 20, 1777–February 23, 1855) intro- duced the Gauss linking integral for computing linking numbers. His student, Johann B. Listing (July 25, 1808–December 24, 1882; Listing was the first to use the term “topology” in 1847), continued the study. Note. In 1877 Scottish physicist Peter Tait published a series of papers on the enumeration of knots (see P.G. -
Arxiv:1708.08132V1 [Math.CO] 27 Aug 2017 Ftevsiivko Nain Hoy[3.Te R Eyueu for Useful Very 26]
TOPOLOGICAL TUTTE POLYNOMIAL SERGEI CHMUTOV Abstract. This is a survey of recent works on topological extensions of the Tutte polynomial. Mathematics Subject Classifications: 05C31, 05C22, 05C10, 57M15, 57M25, 57M30 1. Introduction We survey recent works on topological extensions of the Tutte polynomial. The extensions deal with graphs on surfaces. Graphs embedded into surfaces are the main objects of the topological graph theory. There are several books devoted to this subject [23, 28, 38, 43]. Cellular embeddings of graphs can be described as ribbon graphs. Oriented ribbon graphs appear under different names such as rotation systems [43], maps, fat graphs, cyclic graphs, dessins d’enfants [38]. Since the pioneering paper of L. Heffter in 1891 [31] they occur in various parts of mathematics ranging from graph theory, combinatorics, and topology to representation theory, Galois theory, algebraic geometry, and quantum field theory [13, 30, 35, 38, 45, 46]. For example, ribbon graphs are used to enumerate cells in the cell decomposition of the moduli spaces of complex algebraic curves [30, 45, 38]. The absolute Galois group Aut(Q/Q) faithfully acts on the set of ribbon graphs (see [38] and references therein). Ribbon graphs are the main combinatorial objects of the Vassiliev knot invariant theory [13]. They are very useful for Hamiltonicity of the Cayley graphs [25, 26]. M. Las Vergnas [40] found a generalization of the Tutte polynomial to cellu- larly embedded graphs as an application of his matroid perspectives. Recently his polynomial was extended to not necessarily cellularly embedded graphs [24]. In 2001 B. Bollob´as and O. Riordan [4, 5], motivated by some problems of knot theory, introduced a different generalization of the Tutte polynomial for ribbon graphs. -
By Lindsay Mooradian
Knots by Lindsay Mooradian 1 1 Can you untie this Knot? Knot theory is the study of mathematical knots. A knot is a closed non-intersecting curve that is em- bedded in three dimensions. A break in a knot diagram represents a shadow, as knots are three dimen- sional objects being drawn on a two dimensional plane. The break helps us distinguish what strand is on top and what strand is below. Knot theorists are driven by the question: How do we tell if two knots are the same? 1 1 In a talk given by Candice Price at Harvey Mudd College in 2019, she discusses her research in knotted DNA structures. She uses well defined definitions and reasonings as to why knots are studied and the basics of knot theory. Candice Price (University of San Diego) on “Unravelling Biochemistry Mysteries: Knot theory applied to biochemistry” So, why do we study knots? Knot theory has real world applica- tions as simple as unknoting your headphones and as complex as understanding antibiotics and potentially curing cancer! 1 1 Price’s research involves knot theory because DNA can become knotted just like a mathematical knot. A topoisomerase I and II are proteins that are important for the replication transcription, unknotting, unlinking. When there is a knotted piece of DNA it needs to be untangled in order to replicate and make a new cell. This can cause cell death if the DNA can not become untangled. A type 11 to- poisomerase can come in when DNA is knotted and unlink these two pieces of DNA so it can replicate. -
Using Knot Theory to Model and Analyze Dna Replication and Recombination
USING KNOT THEORY TO MODEL AND ANALYZE DNA REPLICATION AND RECOMBINATION by Megan K. Farrell, B.S. A thesis submitted to the Graduate Council of Texas State University in partial fulfillment of the requirements for the degree of Master of Science with a Major in Mathematics December 2018 Committee Members: David Snyder, Chair Eugene Curtin Anton Dochtermann COPYRIGHT by Megan K. Farrell 2018 FAIR USE AND AUTHOR'S PERMISSION STATEMENT Fair Use This work is protected by the Copyright Laws of the United States (Public Law 94{553, section 107). Consistent with fair use as defined in the Copyright Laws, brief quotations from this material are allowed with proper acknowledgment. Use of this material for financial gain without the author's express written permission is not allowed. Duplication Permission As the copyright holder of this work I, Megan K. Farrell, refuse permission to copy in excess of the \Fair Use" exemption without my written permission. DEDICATION In memory of my mother, who always supported my pursuit of mathematics. ACKNOWLEDGMENTS Thanks to my advisor, Dr. David Snyder, for all the assistance and encouragement he gave me during my mathematical studies. Thanks to my mentor, Dr. Ray Treinen for his support and guidance through my academic career, and for motivating me to continue my mathematical studies. Thanks to Dr. Eugene Curtin for his excellent teaching skills that piqued the interest of an adolescent mathematician. Lastly, thanks to Dr. Anton Dochtermann for his assistance on this committee and for his wonderful instruction in my graduate academics. v TABLE OF CONTENTS Page ACKNOWLEDGEMENTS..............................................v LIST OF FIGURES................................................... -
Knot Theory and Its Applications
Alma Mater Studiorum Universita` degli Studi di Bologna Facolt`adi Scienze Matematiche, Fisiche e Naturali Corso di Laurea in Matematica Tesi di Laurea in Geometria III Knot Theory and its applications Candidato: Relatore: Martina Patone Chiar.ma Prof. Rita Fioresi Anno Accademico 2010/2011 - Sessione II . Contents Introduction 11 1 History of Knot Theory 15 1 The Firsts Discoveries . 15 2 Physic's interest in knot theory . 19 3 The modern knot theory . 24 2 Knot invariants 27 1 Basic concepts . 27 2 Classical knot invariants . 28 2.1 The Reidemeister moves . 29 2.2 The minimum number of crossing points . 30 2.3 The bridge number . 31 2.4 The linking number . 33 2.5 The tricolorability . 35 3 Seifert matrix and its invariants . 38 3.1 Seifert matrix . 38 3.2 The Alexander polynomial . 44 3.3 The Alexander-Conway polynomial . 46 4 The Jones revolution . 50 4.1 Braids theory . 50 4.2 The Jones polynomial . 58 5 The Kauffman polynomial . 63 3 Knot theory: Application 69 1 Knot theory in chemistry . 69 4 Contents 1.1 The Molecular chirality . 69 1.2 Graphs . 73 1.3 Establishing the topological chirality of a molecule . 78 2 Knots and Physics . 83 2.1 The Yang Baxter equation and knots invariants . 84 3 Knots and biology . 86 3.1 Tangles and 4-Plats . 87 3.2 The site-specific recombination . 91 3.3 The tangle method for site-specific recombination . 92 Bibliography 96 List of Figures 1 Wolfagang Haken's gordian knot . 11 1.1 Stamp seal (1700 BC) . -
KNOTS and KNOT GROUPS Herath B
Southern Illinois University Carbondale OpenSIUC Research Papers Graduate School 8-12-2016 KNOTS AND KNOT GROUPS Herath B. Senarathna Southern Illinois University Carbondale, [email protected] Follow this and additional works at: http://opensiuc.lib.siu.edu/gs_rp Recommended Citation Senarathna, Herath B. "KNOTS AND KNOT GROUPS." (Aug 2016). This Article is brought to you for free and open access by the Graduate School at OpenSIUC. It has been accepted for inclusion in Research Papers by an authorized administrator of OpenSIUC. For more information, please contact [email protected]. KNOTS AND KNOT GROUPS by H B M K Hiroshani Senarathna B.S., University of Peradeniya, 2011 A Research Paper Submitted in Partial Fulfillment of the Requirements for the Master of Science Department of Mathematics in the Graduate School Southern Illinois University Carbondale December, 2016 RESEARCH PAPER APPROVAL KNOTS AND KNOT GROUPS By Herath Bandaranayake Mudiyanselage Kasun Hiroshani Senarathna A Research Paper Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the field of Mathematics Approved by: Prof. Michael Sullivan, Chair Prof. H. R. Hughes Prof. Jerzy Kocik Graduate School Southern Illinois University Carbondale August 10, 2016 TABLE OF CONTENTS ListofFigures ...................................... ii Introduction....................................... 1 1 History......................................... 2 1.1 EarlyWorks................................... 2 1.2 Laterwork.................................... 5 1.3