"Knot theory"의 두 판 사이의 차이

수학노트
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The Yang-Baxter equation and invariants of links  Turaev
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==introduction==
  
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* {{수학노트|url=매듭이론_(knot_theory)}}
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* Given a knot and a rational number one can define a closed three-manifold by Dehn surgery
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*  Knot complements and 3-manifolds
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** a knot K is either hyperbolic or a torus knot or a satellite knot
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* [[Reid-Walsh conjecture]]
  
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<h5>Kauffman's principle</h5>
 
  
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==knot diagram==
  
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* projection to two dimensional space
  
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<h5>Knot invariants and quantum groups</h5>
 
  
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* using the Boltzmann weights from the various exactly solvable models, we can discover an infinite series of invariants of knots.
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* so the problem is to find a nice set of Boltzmann weights
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==Kauffman's principle==
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==knot invariants==
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* Alexander-Conway polynomial
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* Jones polynomial
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* Vassiliev invariants
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* define them recursively using the skein relation
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* Reidemeister's theorem is used to prove that they are knot invariants
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* 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.
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* 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.
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* 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==
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* Kauffman bracket
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* colored Jones polynomial
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* [[Hecke algebra]]
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* [[Jones polynomials]] and <math>U_q[\mathfrak{sl}(2)]</math>
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==Knot theory, statistical mechanics and quantum groups==
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* [[Knot theory|Knot Theory]] and Statistical Mechanics
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** http://web.phys.ntu.edu.tw/phystalks/Wu.pdf
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* using the Boltzmann weights from the various exactly solvable models, we can discover an infinite series of invariants of knots
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* 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==
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* [[topological quantum field theory(TQFT)]]
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==knot and QFT==
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* [[knot and quantum field theory]]
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==related items==
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* [[Knot theory and q-series]]
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* [[volume of hyperbolic threefolds and L-values]]
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxUlVqT190VzRTdGs/edit
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==books==
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* Atiyah, Michael The Geometry and Physics of Knots
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==encyclopedia==
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* http://en.wikipedia.org/wiki/knot_theory
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* http://en.wikipedia.org/wiki/List_of_knot_theory_topics
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* [http://en.wikipedia.org/wiki/Link_%28knot_theory%29 http://en.wikipedia.org/wiki/Link_(knot_theory)]
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* http://en.wikipedia.org/wiki/Reidemeister_move
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==articles==
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* [http://dx.doi.org/10.1142/S0217732395001526 A link invariant from quantum dilogarithm]
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** Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418
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* [http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf Knot theory and statistical mechanics]
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** Richard Altendorfer
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* http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf
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* [http://siba2.unile.it/ese/issues/1/19/Notematv9supplp17.pdf Knot and physics]
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**  Kauffman, 1989
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* [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102650387 On knot invariants related to some statistical mechanical models.]
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** V. F. R. Jones, 1989
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* [http://www.kryakin.com/files/Invent_mat_%282_8%29/92/92_05.pdf The Yang-Baxter equation and invariants of links]
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** Turaev, 1988
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* [http://www.bkfc.net/altendor/IntroductionToKnotTheory.pdf An Introduction to Knot Theory]
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** Richard Altendorfer
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==question and answers(Math Overflow)==
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* http://mathoverflow.net/search?q=knot+quantum
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[[분류:math and physics]]
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[[분류:Knot theory]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q849798 Q849798]
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===Spacy 패턴 목록===
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* [{'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


Knot theory, statistical mechanics and quantum groups

  • 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


related items


computational resource


books

  • Atiyah, Michael The Geometry and Physics of Knots


encyclopedia


articles


question and answers(Math Overflow)

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'knot'}, {'LEMMA': 'theory'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Knot_theory&oldid=51863"