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

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BRAIDED KNOTS 2S est Seifert circles not nested <strong>with</strong> the others and sending a point inside this circle to infinity. First let us apply the Vogel algorithm to a very simple knot (the unknot, in fact) to see how ehanges of infinity oeeur (Figure 2.10). Following the first smoothing, there are no eountries in turmoil, and no nested Seifert circles. So we proeeed to the eommand Do change infinity, which must be earried out twice-(b) beeomes (c) and (c) beeomes (d)-to obtain a eoiled knot (d), whieh ean then be unrolled into a braid as before (e). Figure 2.11 shows how Vogel's algorithm eoils a knot <strong>with</strong> five erossings.2 Following the initial smoothing, the 100p (in the eomputational sense of the word) eontains two perestroikas (Figures 2.11 b and 2.1 1e); it is followed by a ehange-of-infinity move. The result (Figure 2.11d) is indeed a eoiled knot, even if it does not 100k like one. To make sure, I have redrawn it twice (Figures 2.12b, 2.12e). The reader will have no diffieulty reeognizing the rolled-up knot from Figure 2.4a (and so ean admire the desired braid, which appeared earlier as Figure 2.4b). (a) (b) Figure 2.10. Vogel algorithm applied to the unknot.
U KNOTI %%%%%%%%%%%%%% (a) (b) (c) (d) Figure 2.1 1. Vogel algorithm applied to knot 52.
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 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: 24 KNOTS Figure 2.9. Perestroika of
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
Page 91 and 92: 70 KNOTS Of course, the reader who
Page 93 and 94: 72 KNOTS particular, he will then s
Page 95 and 96: 74 KNOTS Statistical Models For a g
Page 97 and 98:
76 KNOTS the pIane into two compiem
Page 99 and 100:
78 KNOTS type-B crossings, respecti
Page 101 and 102:
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
Page 118 and 119:
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
Page 124 and 125:
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,
Page 136 and 137:
KNOTS ANO PHYSICS 115 but perhaps n
Page 138 and 139:
KNOTS AND PHYSICS 117 space) is not
Page 140:
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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