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

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I ATOMS ANO KNOTS (Lord Kelvin· 1860) In 1860, the English physicist William Thomson (better known to day by the name of Lord Kelvin, but at the time not yet graced with a noble title) was pondering the fundamental problems linked to the structure of matter. His peers were divided into two enemy camps: those who supported the so-called corpuscular theory, according to which matter is constituted of atoms, rigid little bodies that occupy a precise position in space, and those who felt matter to be a superposi tion of waves dispersed in space-time. Each of these theories provides convincing explanations for certain phenomena but is inadequate for others. Thornson was looking for a way to combine them. And he found one. According to Thomson, matter is indeed consti tuted of atorns. These "vortex atoms," however, are not pointlike ob jects but little knots (see Thomson, 1867). Thus an atom is like a wave that, instead of dispersing in all directions, propagates as a narrow beam bending sharply back on itself-like a snake biting its tail. But this snake could wriggle in a fairly complicated way before biting itself, thus forming a knot (Figure 1.1). The type of knot would then deter mine the physicochemical properties of the atomo According to this 1
2 KNOTS view, molecules are constituted of several intertwined vortex atoms, that is, they are modeled on what mathematicians call links: a set of curves in space that can knot up individually as well as with each other. This theory will no doubt seem rather fanciful to the reader accus tomed to Niels Bohr's planetary model of the atom taught in school. But we are in 1860, the future Nobel laureate will not be bom until25 years later, and the scientific community is taking Thomson's revolu tionary idea seriously. The greatest physicist of the period, James Clerk Maxwell, whose famous equations formed the basis of wave theory, hesitated at first, then warmed to the idea. He insisted that Thomson's theory explained the experimental data accumulated by researchers better than any other. To develop his theory, Thomson needed first of alI to see which dif ferent types of knots are possible; in other words, he had to classify knots. It would then have been possible to classify atoms by associat ing each type of knot with a specific atomo For example, the three Figure 1.1. Model ofan atom?
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 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 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
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50 KNOTS %%%%%%% Digression : The S
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52 KNOTS that n . m = l. (Of course
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54 ) KNOTS Figure 4.5. One knot can
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56 KNOTS of two other nontrivial kn
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SURGERY ANO INVA RIANTS (Conway ·1
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60 KNOTS leaves us with no choice a
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62 Bases: A - adenine T - thymine C
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64 KNOTS Figure 5.5. A supercoiled,
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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
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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
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KNOTS AND PHYSICS (Xxx? . 2004?) Th
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KNOTS ANO PHYSICS 107 ,. Il b· I :
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KNOTS ANO PHYSICS 109 explain the m
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KNOTS ANO PHYSICS 111 Figure 8.2. O
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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
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126 NOTES stractions, which belong
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