Sossinsky:Knots. Mathematics with a twist.pdf - English

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BRAI DED KNOTS 19 Figure 2.4. Unrolling a coiled knot into a braid. ure 2.4 turns around the center C). In this situation, alI that is required to find the braid is to cut the knot along a line extending outward from the center and then to unroll it (Figure 2.4b). But what if the knot is not coiled, for exampIe, the knot shown in Figure 2.Sa? (As readers of the first chapter will know, this knot is called a figure eight knot.) In that case, just move the "fat" part (the thicker line) of the knot (the one "going the wrong way") over the point Con the other si de of the curve. The resulting coiled knot (Figure 2.Sb) can then be unrolled into a braid as in the preceding exampIe (Figure 2.5c). ActualIy, this elegant method (transforming any knot into a coiled knot) is universal, and it alIowed Alexander to prove his theorem. Its weakness-and there is one-is its ineffectiveness from a practical point of view; specificalIy, it is difficult to teach to a computer. Another method of braiding knots, more doabIe and easier to program,
20 KNOTS (a) (b) Figure 2.5. Coiling a figure eight knot and unrolling it into a braid. was invented by the French mathematician Pierre Vogel. The reader not inclined to algorithmic reasoning can blithely skip this description and go on to the study (much simpler and more important) of the group of braids. Vogel's Braiding Algorithm The braiding algorithm transforms any knot into a coiled knot. To describe it I must introduce some definitions related to the planar representations of knots. Assume that a given knot is oriented; that is to say, the direction of the curve (indicated by arrows) has been selected. The planar representation of the knot defines a kind of geographic map in
Page 2 and 3: ALEXEI SOSSINSKY TRANSLATED BY CIS
Page 4: CONTENTS PREFACE vii 1. ATOMS AND K
Page 7 and 8: viii PREFACE This book addresses th
Page 9 and 10: x PREFACE Since Antiquity, the deve
Page 11 and 12: xii PREFACE (a) Figure P.3. Pullies
Page 13 and 14: XIV PREFACE Figure P. S. Links on a
Page 15 and 16: XVI PREFACE The second chapter deal
Page 17 and 18: xviii PREFACE bracket-that is very
Page 20: KNOTS
Page 23 and 24: 2 KNOTS view, molecules are constit
Page 25 and 26: 4 Y : : y o o :: y y: : o : o>-: o
Page 27 and 28: 6 KNOTS Figure 1.5. An altemating k
Page 29 and 30: 8 KNOTS A Mathematical View of Knot
Page 31 and 32: 10 KNOTS here are a few remarks on
Page 33 and 34: 12 KNOTS . Figu re 1.10. A wild kno
Page 35 and 36: 14 KNOTS until then unnoticed, rela
Page 37 and 38: Figure 2.1. Examples of braids. Fig
Page 39: 18 KNOTS is true for the more gener
Page 43 and 44: 22 KNOTS 2.6d, 2.6e). Two Seifert c
Page 45 and 46: 24 KNOTS Figure 2.9. Perestroika of
Page 47 and 48: U KNOTI %%%%%%%%%%%%%% (a) (b) (c)
Page 49 and 50: 28 KNOTS Figure 2.13. The product o
Page 51 and 52: 30 KNOTS 2 3 • n-1 n I I I b,_,
Page 53 and 54: 32 KNOTS (To make this formula easi
Page 55 and 56: 34 KNOTS = bbAAAAAA[bbbbbbBBBBBB]aa
Page 57 and 58: 36 KNOTS Here is his answer: Just p
Page 59 and 60: 38 KNOTS suffice, we will have to s
Page 61 and 62: Figure 3.3. Catastrophes and Reidem
Page 63 and 64: 42 KNOTS Take the first knot and co
Page 65 and 66: 44 KNOTS Figure 3.5. Wolfgang Haken
Page 67 and 68: THE ARITHMETIC OF KNOTS (Schubert·
Page 69 and 70: 48 KNOTS Our immediate goal is to s
Page 71 and 72: 50 KNOTS %%%%%%% Digression : The S
Page 73 and 74: 52 KNOTS that n . m = l. (Of course
Page 75 and 76: 54 ) KNOTS Figure 4.5. One knot can
Page 77 and 78: 56 KNOTS of two other nontrivial kn
Page 79 and 80: SURGERY ANO INVA RIANTS (Conway ·1
Page 81 and 82: 60 KNOTS leaves us with no choice a
Page 83 and 84: 62 Bases: A - adenine T - thymine C
Page 85 and 86: 64 KNOTS Figure 5.5. A supercoiled,
Page 87 and 88: 66 KNOTS For example, the calculati
Page 89 and 90: 68 KNOTS gether). But never mind th
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70 KNOTS Of course, the reader who
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72 KNOTS particular, he will then s
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74 KNOTS Statistical Models For a g
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76 KNOTS the pIane into two compiem
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78 KNOTS type-B crossings, respecti
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80 KNOTS The first ruIe-which, desp
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82 KNOTS same Kauffman bracket. Thi
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84 KNOTS A Little Personal Digressi
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86 KNOTS Let us cali on a classical
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88 KNOTS The two other rules are ob
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FIN ITE-ORDER INVA RIANTS (Vassilie
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FINITE-ORDER INVARIANTS 93 at point
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FINITE-ORDER INVA RIANTS 95 only po
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FIN ITE-ORDER INVARIANTS 97 Slightl
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FINITE-ORDER INVA RIANTS 99 are alI
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FI NITE-ORDER INVA RIANTS 3 3 4 Fig
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FI NITE-ORDER INVA RIANTS 103 The m
Page 126 and 127:
KNOTS AND PHYSICS (Xxx? . 2004?) Th
Page 128 and 129:
KNOTS ANO PHYSICS 107 ,. Il b· I :
Page 130 and 131:
KNOTS ANO PHYSICS 109 explain the m
Page 132 and 133:
KNOTS ANO PHYSICS 111 Figure 8.2. O
Page 134 and 135:
KNOTS AND PHYSICS 113 This theory,
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KNOTS ANO PHYSICS 115 but perhaps n
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KNOTS AND PHYSICS 117 space) is not
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KNOTS AND PHYSICS 119 faces. Is a q
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122 NOTES one of the regions bounde
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124 NOTES Przytycki and Pawel Tracz
Page 147 and 148:
126 NOTES stractions, which belong
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