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

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ATO MS AND KNOTS 13 blindo But actually, it isn't alI that surprising, since almost alI blind mathematicians are (or were) geometers. The spatial intuition that sighted peopie have is based on the image of the world that is pro jected onto their retinas; thus it is a two- (and not three-) dimensional image that is analyzed in the brain of a sighted person. A blind per son's spatial intuition, on the other han d, is primarily the result of tac tile and operational experience. It is also deeper-in the literal as well as the metaphorieal sense. As we Ieave this digression, note that recent biomathematical stud ies (based on work <strong>with</strong> children and adults who were born blind but gained their sight afterward) have shown that the deepest mathemati cal structures, such as topologieal structures, are innate, whereas finer structures, such as linear structures, are acquired. Thus, at first, the blind person who regains his sight do es not distinguish a square from a circle: he sees onIy their topological equivalence. In contrast, he im mediateIy sees that the torus is not a sphere. As for us, our tendency to consider what we see the "absolute truth" often makes us conceive the world in a very flat and superficial way. The Failure ofThomson's Theory While European physicists were debating the merits of Thomson's theory and Tait was filling in his tabies of knots, another researcher unknown and working in an immense, underdeveloped country was, Iike Thomson and Tait, reflecting on the structure of matter. He, too, was trying to estabIish tabies of atoms but, little inclined to geo metrie considerations, he based his tabies on arithmeticai reiation ships among the various properties of chemieal elements. This scientist made an unexpected discovery: there are very simpIe,
14 KNOTS until then unnoticed, relationships among the chemical properties. And he published what today is called the periodic table of elements. It would take some time for this remarkable discovery to be recognized in western Europe. Once that happened, my compatriot Mendeleev buried the idea of atoms as knots. Thomson's theory had not done much for chemistry, and it was quickly replaced by the arithmetical theory of Mendeleev. And physicists, ashamed and embarrassed, for got about knots for nearly a century. It was the mathematicians who would take up the subject once again.
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: 12 KNOTS . Figu re 1.10. A wild kno
Page 37 and 38: Figure 2.1. Examples of braids. Fig
Page 39 and 40: 18 KNOTS is true for the more gener
Page 41 and 42: 20 KNOTS (a) (b) Figure 2.5. Coilin
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
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68 KNOTS gether). But never mind th
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70 KNOTS Of course, the reader who
Page 93 and 94:
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
Page 103 and 104:
82 KNOTS same Kauffman bracket. Thi
Page 105 and 106:
84 KNOTS A Little Personal Digressi
Page 107 and 108:
86 KNOTS Let us cali on a classical
Page 109 and 110:
88 KNOTS The two other rules are ob
Page 111 and 112:
FIN ITE-ORDER INVA RIANTS (Vassilie
Page 114 and 115:
FINITE-ORDER INVARIANTS 93 at point
Page 116 and 117:
FINITE-ORDER INVA RIANTS 95 only po
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FIN ITE-ORDER INVARIANTS 97 Slightl
Page 120 and 121:
FINITE-ORDER INVA RIANTS 99 are alI
Page 122 and 123:
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
Page 143 and 144:
122 NOTES one of the regions bounde
Page 145 and 146:
124 NOTES Przytycki and Pawel Tracz
Page 147 and 148:
126 NOTES stractions, which belong
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