Ohtsuki T. — Quantum invariants: a study of knot, 3-manifolds, and their sets :: Электронная библиотека попечительского совета мехмата МГУ

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Ohtsuki T. — Quantum invariants: a study of knot, 3-manifolds, and their sets

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Название: Quantum invariants: a study of knot, 3-manifolds, and their sets

Автор: Ohtsuki T.

Аннотация:

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik–Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern–Simons field theory and the Wess–Zumino–Witten model are described as the physical background of the invariants.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 451

Добавлена в каталог: 31.10.2010

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Предметный указатель

$C_{d}$ -equivalent      187 380
$C_{d}^{\mathbb{Q}}$ -equivalent      187
$sl_{3}$ linear skein      223
$Y_{d}$ -equivalent      329 391
$Y_{d}^{\mathbb{Q}}$ -equivalent      329
4-manifold according to surgery      219
4T relation      179
6T relation      193
Aarhus integral      301
Adjoint representation      100
Adjoint representation of Lie algebra      100
Affine Lie algebra      429
Alexander polynomial as operator invariant      59
Alexander polynomial derived from quantum group      97
Alexander polynomial via braid representation      37 38
Alexander polynomial via Seifert matrix      21 22
Algebraically split      306
Antipode of a component of a link      63 153
Antipode of Hopf algebra      64
Antipode of Jacobi diagram      136
Arrow diagram      193
AS relation, among Jacobi diagrams      135
AS relation, among oriented Jacobi diagrams      195
AS relation, among tree claspers      386
Associator in      4 111 148 367
Associator in quasi-bialgebra      124 367
Associator with rational coefficients      369
Bialgebra      123
Blackboard framing      15
Box in linear skein      224
Box in sh linear skein      353
Box of graph clasper      377
Box, associated with intertwiner      235
Braid      23
Braid, group      24
Burau representation      36
Burau representation of braid group      36
Casimir element      101
Casson invariant      312
Casson — Walker invariant      257
Casson — Walker — Lescop invariant      284
Charge conservation      29
Chern — Simons functional      406
Chord      134
Chord diagram      134
Chord in homology      3
Chord, diagram      134
Clasper      181
Closure of braid      24
Closure of tangle      58
Comultiplication of a component of a link      63 153
Comultiplication of Hopf algebra      64
Comultiplication of Jacobi diagram      136
Configuration space      103 201
Conformal block      429 430
Conformal field theory      424 435
Conway polynomial      161
Counit of a component of a link      63 153
Counit of Hopf algebra      64
Counit of Jacobi diagram      136
Crossing      3
D-equivalent      187 329 380 391
Degree of Jacobi diagram      134
Degree of tree clasper      380
Degree of Vassiliev invariant      176 181 192
Dehn twist      215
Diagram of framed link      15
Diagram, arrow      193
Drinfel’d associator      111
Drinfel’d associator in $\mathcal{A}\downarrow\downarrow\downarrow$       111 148
Drinfel’d series      366
Dual module      82
Dual representation of Hopf algebra      81
Dual, module      82
Dual, representation      81
Elementary parenthesized braid      121
Elementary tangle diagram oriented      46
Elementary, parenthesized braid      121
Elementary, quasi-tangle diagram      145
Exterior      212
Fenn — Rourke Move      220
Fermat limit      259
Feynman diagram      134
FI relation      143
FI relation, among arrow diagrams      193
FI relation, among chord diagrams      179
Figure-eight knot      2
Finite type invariant of homology      3
Finite type invariant of knots      176
Finite type invariant, spheres      306 312
Formal Knizhnik — Zamolodchikov(KZ) equation      142
Framed      3
Framed Kontsevich invariant      150
Framed Kontsevich invariant.      150
Framed link      15
Framed of      3
Framed of link      15
Framed, manifold      230
Framing      212
Gauss diagram      190
Gaussian integral      264
Gaussian sum      255 260 438
Generalized symplectic quotient      414
Graded weight system      167
Graph chord      394
Graph clasper      394
Group-like      157 366
Handle      218
Handle slide      218 219 2
Handlebody      214
Heegaard splitting      214
Hexagon relation in a space of Jacobi diagrams      148
Hexagon relation in quasi-bialgebra      124
Hexagon relation of Drinfel’d series      366
Hexagon relation of sliced parenthesized braid      120
Homology      3
Hopf algebra      64
IHX relation among graph claspers      397
IHX relation among Jacobi diagrams      135
IHX relation among oriented Jacobi diagrams      195
IHX relation among tree claspers      387
Integrability condition      104
Integral on configuration space      205
Integral surgery      212
Intersection form      227
Intertwiner      82 166
Invariant 2-invariant, tensor      100
Invariant 2-invariant, tensor of $sl_{2} \bigotimes sl_{2}$       102
Invariant 2-invariant, tensor of $sl_{N} \bigotimes sl_{N}$       103
Irreducible module      77
Irreducible representation      77
Irreducible representation of Hopf algebra      77
Isolated chord      179
Isotopic      2 23
Isotopy      2 3
Isotopy invariant      2
Jacobi diagram      134
Jones polynomial as operator invariant      57
Jones polynomial derived from quantum group      93
Jones polynomial via braid representation      31
Jones polynomial via Kauffman bracket      11
Jones — Wenzl idempotent      224
Kauffman bracket      8
Kauffman bracket as invariant of framed links      16
Kauffman bracket as operator invariant      52
Killing form      100
Kirby move      217
Knizhnik — Zamolodchikov(KZ) equation      104 434
Knot diagram      3
Kontsevich invariant      144

KZ equation in Jacobi diagram      142
Leaf (of clasper)      376
Legendre symbol      256
Lens space      241
Link      2
Link diagram      3
Link number      7
Linking matrix      227
LMO invariant      286
Loop algebra      415
Loop group      414 426
Manifold framed      230
Mapping degree on configuration space      205
Markov move      26
Meridian disc      213
Meridian of solid torus      212
Module      77
Monodromy representation      108 109 143
Monodromy representation of braid group      108 109 143
Multi-variable Alexander polynomial      97
Multiple zeta function      373
Negative crossing      7
Ohtsuki invariant      257
Operator invariant of oriented tangle      55
Operator invariant of unoriented tangle      50
Operator invariant via ribbon Hopf algebra      78
Oriented Jacobi diagram      194
Oriented link      5
Parenthesized braid      119
Parenthesized braid in quasi-bialgebra      124
Parenthesized braid of Drinfel’d series      366
Parenthesized braid of sliced parenthesized braid      120
Parenthesized set of dots      118
path integral      407 425
Pentagon relation in a space of Jacobi diagrams      148
Perturbative invariant of 3-manifolds, perturbative PG invariant      259
Perturbative invariant of 3-manifolds, perturbative SO(3)      257 266
Positive and negative crossing      7
Primitive finite type invariant      328
Primitive Jacobi diagram      159 286
Primitive Kontsevich invariant      159
Primitive LMO invariant      286
Primitive, finite type invariants      328
Primitive, Jacobi diagram      159 286
Primitive, Kontsevich invariant      159
Primitive, LMO invariant      286
Primitive, Vassiliev invariant      186
Pure braid      103
Pure braid, group      103
q-binomial coefficient      333
Q-exponential map      87
Quantization      414
Quantized Yang — Baxter equation      67
Quantum dimension      78 90
Quantum group      86
Quantum Hilbert space      408 410
Quantum Hilbert space for surface with boundary      415
Quantum Hilbert space in Chern — Simons theory      412 413
Quantum integer      87
Quantum invariant of 3-manifolds      407
Quantum invariant of 3-manifolds, invariant via linear skein      231
Quantum invariant of 3-manifolds, invariant via quantum invariants of links      240
Quantum invariant of 3-manifolds, quantum G invariant      244
Quantum invariant of 3-manifolds, quantum PG invariant      244
Quantum invariant of 3-manifolds, quantum PSU(3) invariant via linear skein      355
Quantum invariant of 3-manifolds, quantum SU(2) invariant via linear skein      227
Quantum invariant of 3-manifolds, quantum SU(2) invariant via quantum invariants of links      234
Quantum invariant of 3-manifolds, quantum SU(3) invariant via linear skein      354
Quantum invariant of knots and links      63 407
Quantum invariant of knots and links, invariant via quantum group      88
Quantum invariant of knots and links, quantum $(sl_{2}, V)$ invariant via Kauffman bracket      93 458
Quantum invariant of knots and links, quantum $(sl_{2}, V)$ invariant via quantum group      92
Quantum invariant of knots and links, quantum $(sl_{2}, V_{n})$ invariant via linear skein      461
Quantum invariant of knots and links, quantum $(sl_{2}, V_{n})$ invariant via quantum group      90
Quantum invariant of knots and links, quantum $(sl_{2}, \star)$       167
Quantum invariant of knots and links, quantum (g, R) invariant      167
Quantum trace      78
Quasi bialgebra      123
Quasi-bialgebra      123
Quasi-tangle      144
Quasi-triangular homology      65
Quasi-triangular, Hopf algebra      65
Quasi-triangular, quasi-bialgebra      124
R Matrix      29
Rational homology      254
Reidemeister move among Gauss diagrams      191
Reidemeister move among link diagrams of oriented links      6
Reidemeister move among link diagrams of unoriented links      5
Representation of Hopf algebra      77
Representation ring      240
Ribbon homology      71
Ribbon Hopf algebra      71
Schur’s lemma      77
Segal — Sugawara construction      433
Seifert matrix      20
Seifert surface      17
Signature      230
Singular Knizhnik — Zamolodchikov(KZ) equation      176
Singular knot      176
Skein relation      13
Skein relation of Alexander polynomial      39
Skein relation of Jones polynomial      13
Skein relation of Kauffman bracket      9 457
Skein relation of quantum $(sl_{2}, V)$ invariant      92
Skein relation of quantum $(sl_{N}, V)$ invariant      170
Skeleton, 1-skeleton      117 215
Sliced, parenthesized braid      119
Sliced, quasi-tangle diagram      145
Sliced, tangle diagram      44
Solid torus      212
Space of Jacobi diagram      134
Sphere      391
Sphere integral homology      306
Sphere on knot      379
State      79
State, sum      79
STU relation, among Jacobi diagrams      135
STU relation, among oriented Jacobi diagrams      195
STU relation, among tree claspers      386
Surgery      212
Surgery, presentation of 3-manifold      214
Symmetric group      28
Symplectic quotient      411
Tangle      42
Tangle diagram      42
Tangle, 1-tangle      58 197 207
Tangle, 2-tangle      58
Tangle, virtual 1-tangle      197
Temperley — Lieb algebra      223
Tensor module      81
Tensor representation of Hopf algebra      81
Tensor, module      81
Tensor, representation      81
Topological invariant      211
Topological quantum field theory      410
Torus      212
Tree chord      379
Tree clasper      379
Tree clasper in homology      3
Trefoil knot      2
Trivalent diagram      134
Trivalent vertex      134
Trivial knot      2
Truncated q-exponential map      94
Turaev move among oriented sliced diagrams      47
Turaev move among unoriented sliced diagrams      45
Twisting of associator in $\mathcal{A}(\downarrow\downarrow\downarrow)$       368
Twisting of quasi-triangular quasi-bialgebra      126
Uni-trivalent graph      134
Unit module      82

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