Table Of Content

Lin Knot Knot Theory by Computer SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman (Univ. of Illinois, Chicago) The Series on Knots and Everything: is a book series polarized around the theory of knots. Volume 1 in the series is Louis H Kauffman’s Knots and Physics. One purpose of this series is to continue the exploration of many of the themes indicated in Volume 1. These themes reach out beyond knot theory into physics, mathematics, logic, linguistics, philosophy, biology and practical experience. All of these outreaches have relations with knot theory when knot theory is regarded as a pivot or meeting place for apparently separate ideas. Knots act as such a pivotal place. We do not fully understand why this is so. The series represents stages in the exploration of this nexus. Details of the titles in this series to date give a picture of the enterprise. Published: Vol. 1: Knots and Physics (3rd Edition) by L. H. Kauffman Vol. 2: How Surfaces Intersect in Space — An Introduction to Topology (2nd Edition) by J. S. Carter Vol. 3: Quantum Topology edited by L. H. Kauffman & R. A. Baadhio Vol. 4: Gauge Fields, Knots and Gravity by J. Baez & J. P. Muniain Vol. 5: Gems, Computers and Attractors for 3-Manifolds by S. Lins Vol. 6: Knots and Applications edited by L. H. Kauffman Vol. 7: Random Knotting and Linking edited by K. C. Millett & D. W. Sumners Vol. 8: Symmetric Bends: How to Join Two Lengths of Cord by R. E. Miles Vol. 9: Combinatorial Physics by T. Bastin & C. W. Kilmister Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics by W. M. Honig Vol. 11: History and Science of Knots edited by J. C. Turner & P. van de Griend RokTing - Linknot.pmd 2 9/26/2007, 11:47 AM Vol. 12: Relativistic Reality: A Modern View edited by J. D. Edmonds, Jr. Vol. 13: Entropic Spacetime Theory by J. Armel Vol. 14: Diamond — A Paradox Logic by N. S. Hellerstein Vol. 15: Lectures at KNOTS ’96 by S. Suzuki Vol. 16: Delta — A Paradox Logic by N. S. Hellerstein Vol. 17: Hypercomplex Iterations — Distance Estimation and Higher Dimensional Fractals by Y. Dang, L. H. Kauffman & D. Sandin Vol. 19: Ideal Knots by A. Stasiak, V. Katritch & L. H. Kauffman Vol. 20: The Mystery of Knots — Computer Programming for Knot Tabulation by C. N. Aneziris Vol. 21: LINKNOT: Knot Theory by Computer by S. Jablan & R. Sazdanovic Vol. 24: Knots in HELLAS ’98 — Proceedings of the International Conference on Knot Theory and Its Ramifications edited by C. McA Gordon, V. F. R. Jones, L. Kauffman, S. Lambropoulou & J. H. Przytycki Vol. 25: Connections — The Geometric Bridge between Art and Science (2nd Edition) by J. Kappraff Vol. 26: Functorial Knot Theory — Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants by David N. Yetter Vol. 27: Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy by H. Pierre Noyes; edited by J. C. van den Berg Vol. 28: Beyond Measure: A Guided Tour Through Nature, Myth, and Number by J. Kappraff Vol. 29: Quantum Invariants — A Study of Knots, 3-Manifolds, and Their Sets by T. Ohtsuki Vol. 30: Symmetry, Ornament and Modularity by S. V. Jablan Vol. 31: Mindsteps to the Cosmos by G. S. Hawkins Vol. 32: Algebraic Invariants of Links by J. A. Hillman Vol. 33: Energy of Knots and Conformal Geometry by J. O'Hara RokTing - Linknot.pmd 3 9/26/2007, 11:47 AM Vol. 34: Woods Hole Mathematics — Perspectives in Mathematics and Physics edited by N. Tongring & R. C. Penner Vol. 35: BIOS — A Study of Creation by H. Sabelli Vol. 36: Physical and Numerical Models in Knot Theory edited by J. A. Calvo et al. Vol. 37: Geometry, Language, and Strategy by G. H. Thomas Vol. 38: Current Developments in Mathematical Biology edited by K. Mahdavi, R. Culshaw & J. Boucher Vol. 39: Topological Library Part 1: Cobordisms and Their Applications edited by S. P. Novikov and I. A. Taimanov Vol. 40: Intelligence of Low Dimensional Topology 2006 edited by J. Scott Carter et al. Vol. 41: Zero to Infinity: The Fountations of Physics by P. Rowlands RokTing - Linknot.pmd 4 9/26/2007, 11:47 AM K(gE Series on Knots and Everything - Vol. 21 LinKnot Knot Theory by Computer Slavikjablan Radmila Sazdanovic The Mathematical Institute, Belgrade, Serbia >World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Series on Knots and Everything — Vol. 21 LINKNOT Knot Theory by Computer Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-277-223-7 ISBN-10 981-277-223-5 Printed in Singapore. RokTing - Linknot.pmd 1 9/26/2007, 11:47 AM August29,2007 16:40 WorldScientificBook-9inx6in ws-book9x6 Preface Knottheoryis a new andrichfieldof mathematics. Although“real”knots are familiar to everyone and many ideas in knot theory can be formulated in everyday language, it is an area abundant with open questions. One of the main ideas of this book is to avoid obvious classification of knots and links according to their number of components. For this reasonknotsandlinksarereferredtoasKLsandtreatedtogetherwhenever possible. KLsaredenoted by Conwaysymbols,a geometrical-combinatorialway todescribeandderiveKLs. ThesamenotationisusedintheMathematica based computer program LinKnot that represents an integral part of this book. LinKnot is not only a supplementary computer program, but the bestandmostefficienttoolforobtainingalmostalloftheresultspresented in the book, that belong to the field of experimental mathematics. Hands-on computations using Mathematica or the webMathematica package LinKnot along with detailed illustrations facilitate better learn- ing and understanding. The program LinKnot can be downloaded from the web address http://www.mi.sanu.ac.yu/vismath/linknot/ and used as apowerfuleducationalandresearchtoolforexperimentalmathematics–im- plementationofCaudron’sideasandthe Conwaynotationenablesworking with large families of knots and links. The electronic version of this book and the programLinKnot that provideswebMathematica on-line computations are available at the address http://math.ict.edu.yu/. Each knot theory problem described in this book is accompanied with the corresponding LinKnot function that enables the reader to actively use the program LinKnot, not only for illustrating some problems, but for computationsandexperimentation. LinKnotissoftwareopentofuturede- velopment: areadercanchangeitoraddnewfunctions. Forthesystematic v August29,2007 16:40 WorldScientificBook-9inx6in ws-book9x6 vi LinKnot and exhaustive derivation of KLs we have accepted the concept proposed by J.H. Conway and A. Caudron, supported and used in a form adapted for computer implementation. As a prerequisite for the use of the Conway notation,the complete list ofbasic polyhedraup to 20crossingsis givenin the program LinKnot. The key idea is the “vertical” classification of KLs into well-defined categories– worlds, subworlds, classes, and families, according to new sets of recursively computed invariants. Patterns obtained from computing KL invariants imply the existence of more general KL family invariants that agree with all proposed conjectures. We strongly believe that the concept of family invariants will be placed on a firm theoretical founda- tion in the future. New KL tables, organized according to KL families, are given in Appendix A that can be downloaded from the address http://www.mi.sanu.ac.yu/vismath/Appendix.pdf. After a short graph-theoretical introduction, we consider different no- tations for KLs: Gauss, Dowker, and Conway notation, along with their advantages and disadvantages. All basic KL invariants such as the minimum crossing number, minimum writhe, linking number, unknotting or unlinking number, cutting number, and KL properties such as chirality, periodicity,unlinkinggap,andbraidfamilyrepresentativesofKLsaredis- cussed in Chapter 1. InChapter 2 we addresstwo importantproblems: recognitionand gen- eration of KLs. As recognition criteria we consider KL colorings, KL groups,andmorepowerfultoolssuchaspolynomialKLinvariants. Again, we try to show that polynomial KL invariants can be recovered from the Conway notation and recursively computed for KL families. Chapter 3 contains a short excursion into the history of knot theory and places an emphasis on the beauty, universality, and diversity of knot theory through various non-standard applications such as mirror curves, fullerenes, self-referential systems, and KL automata. WewishtothankWolframResearchandICTforsupportingourproject, Professors Mitsuyuki Ochiai and Noriko Imafuji for their cooperation in the developmentofthe programLinKnotandjointdistributionofLinKnot and Knot2000 (K2K), Dror Bar Natan for joining program LinKnot with Mathematica package KnotTheory, and Professors Donald Crowe, Louis Kauffman,JayKappraff,CharlesLivingston,JozefPrzytycki,andThomas Gittings for their advice and suggestions. Slavik Jablan and Radmila Sazdanović August29,2007 16:40 WorldScientificBook-9inx6in ws-book9x6 Contents Preface v 1. Notation of Knots and Links 1 1.1 Basic graph theory . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Shadows of KLs . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 Gauss and Dowker code . . . . . . . . . . . . . . . 16 1.3 KL diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Reidemeister moves. . . . . . . . . . . . . . . . . . . . . . 40 1.5 Conway notation . . . . . . . . . . . . . . . . . . . . . . . 50 1.6 Classification of KLs . . . . . . . . . . . . . . . . . . . . . 59 1.7 LinKnot functions and KL notation . . . . . . . . . . . . 66 1.8 Rational world and KL invariants . . . . . . . . . . . . . 69 1.8.1 Chirality of rational KLs . . . . . . . . . . . . . . 77 1.9 Unlinking number and unlinking gap . . . . . . . . . . . . 81 1.10 Prime and composite KLs . . . . . . . . . . . . . . . . . . 119 1.11 Non-invertible KLs . . . . . . . . . . . . . . . . . . . . . . 125 1.11.1 Tangle types . . . . . . . . . . . . . . . . . . . . . 131 1.11.2 Non-invertible pretzel knots . . . . . . . . . . . . 136 1.11.3 Non-invertible arborescent knots . . . . . . . . . . 140 1.11.4 Non-invertible polyhedral knots . . . . . . . . . . 142 1.12 Reduction of R-tangles . . . . . . . . . . . . . . . . . . . . 145 1.12.1 KLs with unlinking number one . . . . . . . . . . 148 1.13 Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 1.13.1 KLs and braids . . . . . . . . . . . . . . . . . . . 161 1.14 Braid family representatives . . . . . . . . . . . . . . . . . 165 1.14.1 Applications of minimum braids and braid family representatives . . . . . . . . . . . . . . . . . . . . 179 vii