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imported>Pythagoras0 잔글 (찾아 바꾸기 – “* Princeton companion to mathematics(Companion_to_Mathematics.pdf)” 문자열을 “” 문자열로) |
Pythagoras0 (토론 | 기여) |
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==introduction== | ==introduction== | ||
| − | * | + | * {{수학노트|url=매듭이론_(knot_theory)}} |
| − | + | * Given a knot and a rational number one can define a closed three-manifold by Dehn surgery | |
| − | + | * Knot complements and 3-manifolds | |
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| − | Given a knot and a rational number one can define a closed three- | ||
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| − | * Knot complements and 3-manifolds | ||
** a knot K is either hyperbolic or a torus knot or a satellite knot | ** a knot K is either hyperbolic or a torus knot or a satellite knot | ||
| + | * [[Reid-Walsh conjecture]] | ||
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==knot diagram== | ==knot diagram== | ||
| 20번째 줄: | 12번째 줄: | ||
* projection to two dimensional space | * projection to two dimensional space | ||
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==Kauffman's principle== | ==Kauffman's principle== | ||
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* Alexander-Conway polynomial | * Alexander-Conway polynomial | ||
| 41번째 줄: | 31번째 줄: | ||
* This is analogous to studying a physical theory that is in fact relativistic but in which one does not know of a manifestly relativistic formulation - like quantum electrodynamics in the 1930's. | * This is analogous to studying a physical theory that is in fact relativistic but in which one does not know of a manifestly relativistic formulation - like quantum electrodynamics in the 1930's. | ||
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==Jones polynomial== | ==Jones polynomial== | ||
| 50번째 줄: | 40번째 줄: | ||
* colored Jones polynomial | * colored Jones polynomial | ||
* [[Hecke algebra]] | * [[Hecke algebra]] | ||
| − | * [[Jones polynomials]] | + | * [[Jones polynomials]] and <math>U_q[\mathfrak{sl}(2)]</math> |
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==Knot theory, statistical mechanics and quantum groups== | ==Knot theory, statistical mechanics and quantum groups== | ||
| − | * [[Knot theory|Knot Theory]] | + | * [[Knot theory|Knot Theory]] and Statistical Mechanics |
** http://web.phys.ntu.edu.tw/phystalks/Wu.pdf | ** http://web.phys.ntu.edu.tw/phystalks/Wu.pdf | ||
* using the Boltzmann weights from the various exactly solvable models, we can discover an infinite series of invariants of knots | * using the Boltzmann weights from the various exactly solvable models, we can discover an infinite series of invariants of knots | ||
| − | * so the problem is to find a nice set of Boltzmann weights which give non- | + | * so the problem is to find a nice set of Boltzmann weights which give non-trivial invariants |
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==2+1 dimensional TQFT== | ==2+1 dimensional TQFT== | ||
| 72번째 줄: | 58번째 줄: | ||
* [[topological quantum field theory(TQFT)]] | * [[topological quantum field theory(TQFT)]] | ||
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==knot and QFT== | ==knot and QFT== | ||
| 80번째 줄: | 65번째 줄: | ||
* [[knot and quantum field theory]] | * [[knot and quantum field theory]] | ||
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==related items== | ==related items== | ||
| + | * [[Knot theory and q-series]] | ||
| + | * [[volume of hyperbolic threefolds and L-values]] | ||
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| + | ==computational resource== | ||
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxUlVqT190VzRTdGs/edit | ||
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==books== | ==books== | ||
| − | * | + | * Atiyah, Michael The Geometry and Physics of Knots |
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| + | ==encyclopedia== | ||
* http://en.wikipedia.org/wiki/knot_theory | * http://en.wikipedia.org/wiki/knot_theory | ||
* http://en.wikipedia.org/wiki/List_of_knot_theory_topics | * http://en.wikipedia.org/wiki/List_of_knot_theory_topics | ||
* [http://en.wikipedia.org/wiki/Link_%28knot_theory%29 http://en.wikipedia.org/wiki/Link_(knot_theory)] | * [http://en.wikipedia.org/wiki/Link_%28knot_theory%29 http://en.wikipedia.org/wiki/Link_(knot_theory)] | ||
* http://en.wikipedia.org/wiki/Reidemeister_move | * http://en.wikipedia.org/wiki/Reidemeister_move | ||
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==articles== | ==articles== | ||
| − | * [http://dx.doi.org/10.1142/S0217732395001526 A link invariant from quantum dilogarithm] | + | * [http://dx.doi.org/10.1142/S0217732395001526 A link invariant from quantum dilogarithm] |
| − | ** Kashaev, R. M., | + | ** Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418 |
| − | * [http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf Knot theory and statistical mechanics] | + | * [http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf Knot theory and statistical mechanics] |
** Richard Altendorfer | ** Richard Altendorfer | ||
* http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf | * http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf | ||
| − | * [http://siba2.unile.it/ese/issues/1/19/Notematv9supplp17.pdf Knot and physics] | + | * [http://siba2.unile.it/ese/issues/1/19/Notematv9supplp17.pdf Knot and physics] |
| − | ** Kauffman, 1989 | + | ** Kauffman, 1989 |
| − | * [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102650387 On knot invariants related to some statistical mechanical models.] | + | * [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102650387 On knot invariants related to some statistical mechanical models.] |
** V. F. R. Jones, 1989 | ** V. F. R. Jones, 1989 | ||
| − | * [http://www.kryakin.com/files/Invent_mat_%282_8%29/92/92_05.pdf The Yang-Baxter equation and invariants of links] | + | * [http://www.kryakin.com/files/Invent_mat_%282_8%29/92/92_05.pdf The Yang-Baxter equation and invariants of links] |
** Turaev, 1988 | ** Turaev, 1988 | ||
| − | * [http://www.bkfc.net/altendor/IntroductionToKnotTheory.pdf An Introduction to Knot Theory] | + | * [http://www.bkfc.net/altendor/IntroductionToKnotTheory.pdf An Introduction to Knot Theory] |
** Richard Altendorfer | ** Richard Altendorfer | ||
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| − | + | ==question and answers(Math Overflow)== | |
| − | = | + | * http://mathoverflow.net/search?q=knot+quantum |
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| − | + | [[분류:math and physics]] | |
| + | [[분류:Knot theory]] | ||
| + | [[분류:migrate]] | ||
| − | == | + | ==메타데이터== |
| − | [[ | + | ===위키데이터=== |
| + | * ID : [https://www.wikidata.org/wiki/Q849798 Q849798] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'knot'}, {'LEMMA': 'theory'}] | ||
2021년 2월 17일 (수) 02:57 기준 최신판
introduction
- 틀:수학노트
- Given a knot and a rational number one can define a closed three-manifold by Dehn surgery
- Knot complements and 3-manifolds
- a knot K is either hyperbolic or a torus knot or a satellite knot
- Reid-Walsh conjecture
knot diagram
- projection to two dimensional space
Kauffman's principle
knot invariants
- Alexander-Conway polynomial
- Jones polynomial
- Vassiliev invariants
- define them recursively using the skein relation
- Reidemeister's theorem is used to prove that they are knot invariants
- The puzzle on the mathematical side was that these objects are invariants of a three dimensional situation, but one did not have an intrinsically three dimensional definition.
- There were many elegant definitions of the knot polynomials, but they all involved looking in some way at a two dimensional projection or slicing of the knot, giving a two dimensional algorithm for computation, and proving that the result is independent of the chosen projection.
- This is analogous to studying a physical theory that is in fact relativistic but in which one does not know of a manifestly relativistic formulation - like quantum electrodynamics in the 1930's.
Jones polynomial
- Kauffman bracket
- colored Jones polynomial
- Hecke algebra
- Jones polynomials and \(U_q[\mathfrak{sl}(2)]\)
Knot theory, statistical mechanics and quantum groups
- Knot Theory and Statistical Mechanics
- using the Boltzmann weights from the various exactly solvable models, we can discover an infinite series of invariants of knots
- so the problem is to find a nice set of Boltzmann weights which give non-trivial invariants
2+1 dimensional TQFT
knot and QFT
computational resource
books
- Atiyah, Michael The Geometry and Physics of Knots
encyclopedia
- http://en.wikipedia.org/wiki/knot_theory
- http://en.wikipedia.org/wiki/List_of_knot_theory_topics
- http://en.wikipedia.org/wiki/Link_(knot_theory)
- http://en.wikipedia.org/wiki/Reidemeister_move
articles
- A link invariant from quantum dilogarithm
- Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418
- Knot theory and statistical mechanics
- Richard Altendorfer
- http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf
- Knot and physics
- Kauffman, 1989
- On knot invariants related to some statistical mechanical models.
- V. F. R. Jones, 1989
- The Yang-Baxter equation and invariants of links
- Turaev, 1988
- An Introduction to Knot Theory
- Richard Altendorfer
question and answers(Math Overflow)
메타데이터
위키데이터
- ID : Q849798
Spacy 패턴 목록
- [{'LOWER': 'knot'}, {'LEMMA': 'theory'}]