Table Of ContentKNOT
PROJECTIONS
KNOT
PROJECTIONS
Noboru Ito
The University of Tokyo
Meguro-ku, Tokyo, Japan
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Contents
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
