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

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ATO MS ANO KNOTS Figure 1.2. Three knots: the trefoil, the figure eight, and the unknot. knots represented in Figure 1.2, the trefoil, the figure eight, and the trivial knot or unknot, could be models of carbon, oxygen, and hydro gen, respectively. So in the beginning the problem was mathematical (rather than physicochemical): the problem of classifying knots. And it was a Scot tish physicist and mathematician, a friend of Thomson, Peter Guthrie Tait, who set out to solve it. Tait, Kirkman, and the First Tables of Knots According to Tait, a knot, being a closed curve in space, could be represented by a planar curve obtained by projecting it perpendicularly on the horizontal pIane. This projection could have crossings (Figure 1.3), where the projection of one part of the curve crossed an other; the planar representation shows the position in space of the two strands that cross each other by interrupting the Hne that represents the lower strand at the crossing. I have already used this natural way of drawing knots (Figure 1.2) and will continue to use it. Posing the question of how knots should be classified requires 3
4 Y : : y o o :: y y: : o : o>-: o I I l ' I I I I I I I_Y---'" CI .. .. ..-: : .... ::-:--: ://: ...- :_y: o 0 ( :_: ",l oo o KNOTS Figure 1.3. Planar projection of a knot. specifying which knots belong to the same class. It requires, in other words, precisely defining the equivalence of knots. Sut we will leave this definition (the ambient isotopy of knots) for later, limiting ourselves here to an intuitive description. Imagine that the curve defining the knot is a fine thread, flexible and elastic, that can be twisted and moved in a continuous way in space (cutting and gluing back is not al lowed). AlI possible positions will thus be those of the same knot. Changing the position of the curve that defines a knot in space by moving it in a continuous way (without ever cutting or retying it) al ways results in the same knot by definition, but its planar representa tion may become unrecognizable. In particular, the number of cross ings may change. Nevertheless, the natural approach to classifying
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: 2 KNOTS view, molecules are constit
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
Page 71 and 72: 50 KNOTS %%%%%%% Digression : The S
Page 73 and 74: 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
Page 79 and 80:
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
Page 126 and 127:
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
Page 147 and 148:
126 NOTES stractions, which belong
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