Knot projections PDF
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KNOT PROJECTIONS KNOT PROJECTIONS Noboru Ito The University of Tokyo Meguro-ku, Tokyo, Japan CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20160916 International Standard Book Number-13: 978-1-4987-3675-6 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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Contents hapter C 1 (cid:4) Knots, knot diagrams, and knot projections 1 1.1 DEFINITION OFKNOTSFORHIGHSCHOOLSTUDENTS 1 1.2 HOWTODEFINE THENOTIONOFAKNOT DIAGRAM? 2 1.3 KNOTPROJECTIONS 4 1.4 TIPS: DEFINITION OF KNOTS FOR UNDERGRADUATE STUDENTS 4 hapter C 2 (cid:4) Mathematical background (1920s) 7 2.1 REIDEMEISTER’STHEOREMFORKNOTDIAGRAMSAND KNOTPROJECTIONS 8 2.2 PROOFOFREIDEMEISTER’STHEOREMFORKNOT DIAGRAMS 9 2.3 PROOFOFREIDEMEISTER’STHEOREMFORKNOT PROJECTIONS 11 2.4 EXERCISES 13 hapter C 3 (cid:4) Topologicalinvariantofknotprojections(1930s) 17 3.1 ROTATIONNUMBER 17 3.2 CLASSIFICATION THEOREM FOR KNOT PROJECTIONS UNDER THE EQUIVALENCE RELATION GENERATED BY ∆,RII,AND RIII 19 hapter C 4 (cid:4) Classification of knot projections under RI and RII (1990s) 29 4.1 KHOVANOV’SCLASSIFICATION THEOREM 30 4.2 PROOF OF THE CLASSIFICATION THEOREM UNDER RI AND RII 31 4.3 CLASSIFICATIONTHEOREMUNDERRIANDSTRONG RII ORRIAND WEAK RII 37 vii viii (cid:4) Contents 4.4 CIRCLENUMBERSFORCLASSIFICATIONUNDERRIAND STRONG RII 39 4.5 EFFECTIVEAPPLICATIONS OFTHECIRCLENUMBER 44 4.6 FURTHERTOPICS 49 4.7 OPENPROBLEMS AND EXERCISE 50 hapter C 5 (cid:4) ClassificationbyRIandstrongorweakRIII(1996– 2015) 53 5.1 ANEXAMPLE BYHAGGEAND YAZINSKI 53 5.2 VIRO’SSTRONGAND WEAK RIII 55 5.3 WHICHKNOTPROJECTIONSTRIVIALIZEUNDERRIAND WEAK RIII? 57 5.4 WHICHKNOTPROJECTIONSTRIVIALIZEUNDERRIAND STRONG RIII? 61 5.5 OPENPROBLEMAND EXERCISES 70 hapter C 6 (cid:4) Techniques for counting sub chord diagrams (2015–Future) 73 6.1 CHORDDIAGRAMS 73 6.2 INVARIANTS BYCOUNTING SUB CHORDDIAGRAMS 75 6.2.1 Invariant X 75 6.2.2 Invariant H 76 6.2.3 Invariant λ 80 6.3 APPLICATIONS OFINVARIANTS 82 6.4 BASEDARROWDIAGRAMS 84 6.5 TRIVIALIZING NUMBER 86 6.6 EXERCISES 92 hapter C 7 (cid:4) Hagge–Yazinski Theorem (Necessity of RII) 95 7.1 HAGGE–YAZINSKI THEOREMSHOWING THE NON TRIVIALITY OFTHEEQUIVALENCECLASSES OFKNOT PROJECTIONSUNDER RIAND RIII 96 7.1.1 Preliminary 96 7.1.2 Box 97 7.1.3 Structure of the induction 97 7.1.4 Moves 1a, 1b, and RIII inside a rectangle 98 7.1.5 Moves 1a, 1b, and RIII outside the rectangles 98 Contents (cid:4) ix 7.1.5.1 1a 98 7.1.5.2 1b 99 7.1.5.3 RIII 99 7.2 ARNOLD INVARIANTS 100 7.3 EXERCISES 104 hapter C 8 (cid:4) Further result of strong (1, 3) homotopy 107 8.1 STATEMENT 107 8.2 PROOFOFTHESTATEMENT 108 8.3 OPENPROBLEMS AND EXERCISE 117 hapter C 9 (cid:4) Half twisted splice operations,reductivities, unavoidable sets,triplechords, andstrong(1,2) homotopy 121 9.1 SPLICES 122 9.2 ITO SHIMIZU’S THEOREM FOR HALF TWISTED SPLICE OPERATIONS 124 9.3 UNAVOIDABLESETS 125 9.4 UPPER BOUNDSOFREDUCTIVITIES 130 9.5 KNOTPROJECTIONWITHREDUCTIVITYONE: REVISITED 138 9.6 KNOTPROJECTIONWITHREDUCTIVITYTWO 146 9.7 TIPS 150 9.7.1 Tip I 150 9.7.2 Tip II 150 9.8 OPENPROBLEMS AND EXERCISES 151 hapter C 10(cid:4) Weak (1, 2, 3) homotopy 155 10.1 DEFINITIONS 156 10.2 STRONG(1, 2,3) HOMOTOPYAND THEOTHERTRIPLES 156 10.3 DEFINITIONOFTHEFIRSTINVARIANTUNDERWEAK(1, 2,3) HOMOTOPY 158 10.4 PROPERTIESOFW(P) 161 10.5 TIPS 164 10.5.1 The trivializing number is even 164
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