An Excursion in Diagrammatic Algebra Turning a sphere from red to blue 8315.9789814374491-tp.indd 1 10/3/11 12:10 PM 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 *The complete list of the published volumes in the series, can also be found at http://www.worldscibooks.com/series/skae_series.shtml Knots and Everything_original.pmd 1 9/14/2011, 4:14 PM 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. 18: The Self-Evolving Cosmos: A Phenomenological Approach to Nature’s Unity-in-Diversity by S. M. Rosen 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. 22: The Mathematics of Harmony — From Euclid to Contemporary Mathematics and Computer Science by A. Stakhov (assisted by S. Olsen) Vol. 23: Diamond: A Paradox Logic (2nd Edition) by N. S. Hellerstein 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 Knots and Everything_original.pmd 2 9/14/2011, 4:14 PM 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 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 & 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 Vol. 42: The Origin of Discrete Particles by T. Bastin & C. Kilmister Vol. 43: The Holographic Anthropic Multiverse by R. L. Amoroso & E. A. Ranscher Vol. 44: Topological Library Part 2: Characteristic Classes and Smooth Structures on Manifolds edited by S. P. Novikov & I. A. Taimanov Vol. 45: Orbiting the Moons of Pluto Complex Solutions to the Einstein, Maxwell, Schrödinger and Dirac Equations by E. A. Rauscher & R. L. Amoroso Vol. 46: Introductory Lectures on Knot Theory edited by L. H. Kauffman, S. Lambropoulou, S. Jablan & J. H. Przytycki Vol. 47: Introduction to the Anisotropic Geometrodynamics by S. Siparov Vol. 48: An Excursion in Diagrammatic Algebra: Turning a Sphere from Red to Blue by J. S. Carter Knots and Everything_original.pmd 3 9/14/2011, 4:14 PM K E Series on Knots and Everything — Vol. 48 An Excursion in Diagrammatic Algebra Turning a sphere from red to blue J Scott Carter University of South Alabama, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8315.9789814374491-tp.indd 2 10/3/11 12:10 PM 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 Library of Congress Cataloging-in-Publication Data Carter, J. Scott. An excursion in diagrammatic algebra : turning a sphere from red to blue / by J. Scott Carter. p. cm. -- (Series on knots and everything ; v. 48) Includes bibliographical references and index. ISBN-13: 978-981-4374-49-1 (hardcover : alk. paper) ISBN-10: 981-4374-49-0 (hardcover : alk. paper) 1. Low-dimensional topology. I. Title. QA612.14.C37 2012 514'.2--dc23 2011034153 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2012 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. Printed in Singapore. EH - An Excursion in Diagrammatic.pmd 1 10/6/2011, 4:29 PM September7,2011 10:37 WorldScientificBook-9inx6in Carter˙Red˙to˙Blue Dedicated to Tom Banchoff, George Francis, and Tony Robbin: the masters from whom I learned this art v September7,2011 10:37 WorldScientificBook-9inx6in Carter˙Red˙to˙Blue TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk September7,2011 10:37 WorldScientificBook-9inx6in Carter˙Red˙to˙Blue Preface A 2-dimensional sphere can be turned inside out via a process that allows the sphere to pass through itself, but that keeps tangencies intact. Here I turn the sphere from red to blue, or more precisely from magenta to cyan, butIcouldreversethecolorsandturneditfrombluetored,orfromorange togreen. Thecolorchoicesareimmaterial; thisbookdescribestheprocess. The result described in the first sentence is over 50 years old. It gives rise to some of the most complicated yet beautiful examples in modern topology. Eventhestatementseemspeculiar: Iwillhavetodescribeplanes oftangency;Iwillhavetodescribehowtangentplanescanbecomesingular; andIwillhavetoexplaintheprecisenatureofthesetsonwhichthesphere intersects. Please be patient. There are several key ideas to be developed: surface, sphere, tangent planes, singularities of maps, cusps, folds, and intersectionpointsareamongthemainones. Fortunately,alloftheseideas can be seen within the world of experience. Thus, this book is my attempt to explain this example to a lay-public. The example is chosen because of its intrinsic beauty. My version of the sphereeversionbuildsontheworkofothers—principallyuponthesphere eversion of Froisart and Morin. As I develop the narrative below, I will try to motivate the process and explain the key steps in this construction. Within this preface, I will give an overview of the process. As I write this, I imagine you to be a college student beginning a pur- suit of mathematics or science. You might also be a practicing scientist, computer scientist, or engineer who always appreciated mathematics but did not study it beyond the level of calculus. I imagine you to be curious andtobeinvigoratedbymathematicalandscientificimagery. Yourinterest in mathematics might be akin to my own interest in biology or chemistry. You have an appreciation for the aesthetic of the subject but you have not vii