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O`Hara J. — Energy of knots and conformal geometry
O`Hara J. — Energy of knots and conformal geometry

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Название: Energy of knots and conformal geometry

Автор: O`Hara J.

Аннотация:

Energy of knots is a theory that was introduced to create a "canonical configuration" of a knot — a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a "canonical configuration" of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 290

Добавлена в каталог: 31.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$Lk({K}_{1},{K}_{2})$      244
$\alpha$-energy      20 28
$\alpha$-energy polynomial      79
$\Box (K)$      124
$\langle c\rangle(f,g)$      249
$\succ$      58
$\triangle(K)$      124
$\varepsilon$-self avoiding voltage      10
${e}_{*}^{*}$      see “Energy”
${I}_{tv}$      255
${r}^{-\alpha}$-modified $\varepsilon$-off diagonal electrostatic energy      15
${r}^{-\alpha}$-modified $\varepsilon$-self avoiding voltage      15
${r}^{-\alpha}$-modified electrostatic energy      22
${V}_{*}^{*}$      see “Voltage”
${\hat{v}}_{x}(y)$      40 41
${\mathbb R}^{n+1,1}$      136
${\theta}_{k}$      39
${{\mathcal C}c}^{(2,2)}(K)$      180
${{\mathcal C}c}^{(4)}(K)$      184
4-tuple map, extended      186
4-tuple map, of ${S}^{3}$      148
4-tuple map, of a knot      178
Absolute circular self-linking number a la Gauss      254
Admissible polygonal knot      75
Admissible solid torus      75
Annulus, cross-separating annulus      158
Annulus, Lorentzian modulus of      156
Annulus, maximal cross-separating annulus      158
Annulus, maximal non-trivial      232
Annulus, maximal non-trivial, for links      241
Annulus, modulus of      156
Annulus, non-trivial      232
Annulus, non-trivial, for links      241
Area of the twice tangent spheres      223
Argument of the infinitesimal cross ratio      204
Asymptotic crossing number      261
Atwt(K)      223
Average crossing number      249
Base, circls      177 180 183 184
Base, sphere (circle, or points)      152
BASIC      7
Beads radius      131
beta function      79
Brylinski's beta function      see “Beta function”
Canonical symplectic form of a cotangent bundle      191
Captured by a polygonal knot      75
Center of a pencil      152
Charge      5
Circular Gauss map      253
Combined angle      60
Complete family of N,r-polygonal knots      76
Complete system of solid tori      75
Concircular, set of (2,2)-points      180
Concircular, set of generalized, points      186
Concircular, set of osculating, points      184
Concircular, set of, points in ${S}^{3}$      147
Concircular, set of, points of a knot      177
Configuration space      147
Conformal, $(1 - \cos \theta)$ energy      40 208
Conformal, $1 — \cos X$-energy      59
Conformal, $\sin \theta$ energy      213
Conformal, ${(1 — \cos \theta)}^{k}$ energy      61
Conformal, angle      41 204
Conformal, angle ${\theta}_{K}$      39
Conformal, diffeomorphism      138
Conformal, group      136
Conformal, modulus of a solid torus      260
Conformal, sin X-energy      59
Conformal, transformation      35 136
Conformal, transportation      41
Conformal, unit — transportation      40 173
Conformally transported unit vector      173
Conformally transported, unit vector      40
Conformally transported, vector      41
Cosine formula      39
Counter term      see “Knot energy functional”
Cross ratio of four ordered points      149
Cross-separating, annulus      158
Cross-separating, sphere      158
Cross-strand      232
Curvature      see “Geodesic curvature”
De Sitter space      137
Distortion      28
Doubly critical self distance, essential      131
Elastica      263
Energy minimizer      6
Energy, $(\alpha,p)$-energy functional      67
Energy, $\alpha$-energy      20 28
Energy, $\alpha$-energy polynomial      79
Energy, $\varepsilon$-diagonal electrostatic energy      11
Energy, ${E}^{(1)}$      12
Energy, ${E}^{(2)}$      28
Energy, ${E}^{(2)}_{0}$      29 35 204
Energy, ${E}^{(\aplha )}$      20
Energy, ${E}^{(\aplha )}_{\varepsilon}$      15
Energy, ${E}^{\aplha ,p}$      64
Energy, ${E}_{(\zeta\rangle}$      46
Energy, ${E}_{\rm mnts}(K)$      225
Energy, ${E}_{\sin\theta}$      213
Energy, ${E}_{{(1 - \cos\theta)}^{k}}(M)$      61
Energy, ${r}^{-\alpha}$-modified $\varepsilon$-off diagonal electrostatic energy      15
Energy, ${r}^{-\alpha}$-modified electrostatic energy      22
Energy, beta function      79
Energy, conformal $(1 — \cos\theta)$ energy      208
Energy, conformal $\sin\theta$ energy      213
Energy, Eapm      101
Energy, imaginary cross ratio energy      213
Energy, Moebius energy      29
Energy, mutual conformal $(1 — \cos\theta)$ energy      209
Energy, mutual conformal $\sin\theta$ energy      222
Energy, renormalization of electrostatic energy      12
Energy, scale invariant $(\alpha,p)$ energy      114
Energy, unit density $(\alpha,p)$-energy      101
Energy, unit quantity $(\alpha,p)$ energy      114
Energy, X-energy      59
ess.dcsd(h)      131
ess.sd(h)      131
Essential doubly critical self distance      131
Essential self distance      131
Finiteness of knot types      7
Gauss, absolute circular self-linking number a la Gauss      254
Gauss, circular Gauss map      253
Gauss, linking number formula      243
Geodesic curvature      154
GI(f,g)      244
Global radius of curvature, at a point      121
Global radius of curvature, minimum      124
Gromov's distortion      see “Distortion”
Ideal knot      118
Imaginary cross ratio energy      213
INDEX      20 64 101
Infinitesimal cross ratio, argument of      204
Infinitesimal cross ratio, of a knot      203
Infinitesimal cross ratio, of the complex plane      189
Inflation radius      118
Intersecting (two spheres)      154
Inversion in a sphere      34
Inverted open knot      37
Knot energy functional, counter term      13 19 20
Knot energy functional, principal term      20
Knot energy functional, with respect to the ${C}^{2}$-topology      5
L-orthogonal      136
Light-cone      136
Light-like      136
Linking number, absolute circular self-linking number a la Gauss      254
Linking number, Gauss formula      243
Linking number, GI(f,g)      244
Linking number, self-linking number      246
Lorentz group      111 136
Lorentzian, exterior product      143
Lorentzian, modulus of an annulus      156
Lorentzian, modulus of four points      158
Lorentzian, pseudo inner product      110 135
Lorentzian, quadratic form      135
Maximal, cross-separating annulus      158
Maximal, non-trivial annulus      232
Maximal, non-trivial annulus for links      241
Measure of non-trivial spheres      225
Minimum distance      128
Minkowski space      136
Modulus, conformal, of a solid torus      see “Conformal modulus”
Modulus, Lorentzian      see “Lorentzian”
Modulus, of a knot      232
Modulus, of a link      241
Modulus, of an annulus      156
Modulus, of four ordered points      158
Moebius, energy      29
Moebius, group      138
Moebius, transformation      34 138
Mutual, conformal $(1 — \cos \theta)$ energy      209
Mutual, conformal $\sin \theta$ energy      222
Nested (two spheres)      154
Non-trivial, annulus      232
Non-trivial, annulus for links      241
Non-trivial, sphere      175
Non-trivial, sphere in the strict sense      175
Normal projection energy symmetric      60
Optimal embedding      7
Osculating, circle      119
Osculating, sphere      119
Osculating, sphere map      186
Pencil, base sphere (circle, or points) of      152
Pencil, center of      152
Pencil, Poncelet      152
Poncelet, pencil      152
Poncelet, points      153
Principal part      see “Knot energy functional”
Pull-tight      7
Renormalization, electrostatic energy      12
Renormalization, of ${r}^{-\alpha}$-modified electrostatic energy      20
Renormalization, of ${r}^{-\alpha}$-modified voltage      19
Residue formula      57
Rl(K)      118
Ropelength      118
Scale invariant $(\alpha,p)$ energy      114
Self, ${C}^{2}$ self-repulsive      5
Self, essential self-distance      131
Self, linking number      246
Set of concircular points      see “Concircular”
Sl(f)      246
Space-like      136
Sphere, area of the twice tangent spheres      223
Sphere, cross-separating sphere      158
Sphere, inversion in a sphere      34
Sphere, measure of non-trivial spheres      225
Sphere, non-trivial sphere      175
Sphere, non-trivial sphere in the strict sense      175
Sphere, osculating      119
Sphere, twice tangent      173
Sphere, twice tangent sphere      119
Sphere, twice tangent, of a knot      179
Strand, cross-strand      232
Strand, minimal cross-strand      232
Symmetric normal projection energy      60
Thickness      117
Thk(K)      118
Tight      7
Time-like      136
Total variation of the conformally transported unit tangent vectors      255
transportation      see “(unit) conformal transportation”
Twice tangent sphere      119
Twice tangent sphere map      181
Twice tangent sphere, of a knot      179
Unit, density $(\alpha,p)$-energy      101
Unit, quantity $(\alpha,p)$ energy      114
Voltage, $\varepsilon$-self avoiding voltage      10
Voltage, ${r}^{-\alpha}$-modified $\varepsilon$-self avoiding voltage      15
Voltage, ${V}^{(\alpha)}$      19
Voltage, ${V}^{(\alpha)}_{\varepsilon}$      15
Wasted length of an inverted open knot      39
Wr(f)      246
Writhe      246
Writhe, direction      246
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