Knot Projections

Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Contents -- Chapter 1 Knots, knot diagrams, and knot projections -- 1.1 DEFINITION OF KNOTS FOR HIGH SCHOOL STUDENTS -- 1.2 HOW TO DEFINE THE NOTION OF A KNOT DIAGRAM? -- 1.3 KNOT PROJECTIONS -- 1.4 TIPS: DEFINITION OF KNOTS FOR UNDERGRADUATE STUDENTS -- Chapter 2 Mathematical background (1920s) -- 2.1 REIDEMEISTER'S THEOREM FOR KNOT DIAGRAMS AND KNOT PROJECTIONS -- 2.2 PROOF OF REIDEMEISTER'S THEOREM FOR KNOT DIAGRAMS

2.3 PROOF OF REIDEMEISTER'S THEOREM FOR KNOT PROJECTIONS -- 2.4 EXERCISES -- Chapter 3 Topological invariant of knot projections (1930s) -- 3.1 ROTATION NUMBER -- 3.2 CLASSIFICATION THEOREM FOR KNOT PROJECTIONS UNDER THE EQUIVALENCE RELATION GENERATED BY (Omitted), RII, AND RIII -- Chapter 4 Classification of knot projections under RI and RII (1990s) -- 4.1 KHOVANOV'S CLASSIFICATION THEOREM -- 4.2 PROOF OF THE CLASSIFICATION THEOREM UNDER RI AND RII -- 4.3 CLASSIFICATION THEOREM UNDER RI AND STRONG RII OR RI AND WEAK RII

4.4 CIRCLE NUMBERS FOR CLASSIFICATION UNDER RI AND STRONG RII -- 4.5 EFFECTIVE APPLICATIONS OF THE CIRCLE NUMBER -- 4.6 FURTHER TOPICS -- 4.7 OPEN PROBLEMS AND EXERCISE -- Chapter 5 Classification by RI and strong orweak RII (1996- 2015) -- 5.1 AN EXAMPLE BY HAGGE AND YAZINSKI -- 5.2 VIRO'S STRONG AND WEAK RIII -- 5.3 WHICH KNOT PROJECTIONS TRIVIALIZE UNDER RI AND WEAK RIII? -- 5.4 WHICH KNOT PROJECTIONS TRIVIALIZE UNDER RI AND STRONG RIII? -- 5.5 OPEN PROBLEM AND EXERCISES

Chapter 6 Techniques for counting sub chord diagrams (2015-Future) -- 6.1 CHORD DIAGRAMS -- 6.2 INVARIANTS BY COUNTING SUB-CHORD DIAGRAMS -- 6.2.1 Invariant X -- 6.2.2 Invariant H -- 6.2.3 Invariant? -- 6.3 APPLICATIONS OF INVARIANTS -- 6.4 BASED ARROW DIAGRAMS -- 6.5 TRIVIALIZING NUMBER -- 6.6 EXERCISES -- Chapter 7 Hagge-Yazinski Theorem (Necessity of RI)

7.1 HAGGE-YAZINSKI THEOREM SHOWING THE NON TRIVIALITY OF THE EQUIVALENCE CLASSES OF KNOT PROJECTIONS UNDER RI AND RIII -- 7.1.1 Preliminary -- 7.1.2 Box -- 7.1.3 Structure of the induction -- 7.1.4 Moves 1a, 1b, and RIII inside a rectangle -- 7.1.5 Moves 1a, 1b, and RIII outside the rectangles -- 7.1.5.1 1a -- 7.1.5.2 1b -- 7.1.5.3 RIII -- 7.2 ARNOLD INVARIANTS -- 7.3 EXERCISES -- Chapter 8 Further result of strong (1, 3) homotopy -- 8.1 STATEMENT -- 8.2 PROOF OF THE STATEMENT -- 8.3 OPEN PROBLEMS AND EXERCISE

Chapter 9 Half-twisted splice operations, reductivities, unavoidable sets, triple chords, and strong (1, 2) homotopy

IMD-Felder maschinell generiert