"Knot theory"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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* Knot complements and 3-manifolds | * 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== | ||
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* projection to two dimensional space | * projection to two dimensional space | ||
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==Kauffman's principle== | ==Kauffman's principle== | ||
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* [[Hecke algebra]] | * [[Hecke algebra]] | ||
* [[Jones polynomials]] and <math>U_q[\mathfrak{sl}(2)]</math> | * [[Jones polynomials]] and <math>U_q[\mathfrak{sl}(2)]</math> | ||
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* so the problem is to find a nice set of Boltzmann weights which give non-trivial invariants | * 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== | ||
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* [[knot and quantum field theory]] | * [[knot and quantum field theory]] | ||
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==related items== | ==related items== | ||
| − | * [[volume of hyperbolic threefolds | + | * [[volume of hyperbolic threefolds and L-values]] |
| 117번째 줄: | 78번째 줄: | ||
==books== | ==books== | ||
| − | * | + | * Atiyah, Michael The Geometry and Physics of Knots |
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| 128번째 줄: | 88번째 줄: | ||
* http://en.wikipedia.org/wiki/Reidemeister_move | * http://en.wikipedia.org/wiki/Reidemeister_move | ||
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==articles== | ==articles== | ||
| 167번째 줄: | 108번째 줄: | ||
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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]] | [[분류:math and physics]] | ||
2013년 6월 24일 (월) 05:25 판
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
- 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)