"Knot theory"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| 2번째 줄: | 2번째 줄: | ||
* {{수학노트|url=매듭이론_(knot_theory)}} | * {{수학노트|url=매듭이론_(knot_theory)}} | ||
| − | * Given a knot and a rational number one can define a closed three- | + | * Given a knot and a rational number one can define a closed three-manifold by Dehn surgery |
* 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 | ||
| 16번째 줄: | 16번째 줄: | ||
==Kauffman's principle== | ==Kauffman's principle== | ||
| − | + | ||
| − | + | ||
| − | == | + | ==knot invariants== |
* Alexander-Conway polynomial | * Alexander-Conway polynomial | ||
| 31번째 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. | ||
| − | + | ||
| − | + | ||
==Jones polynomial== | ==Jones polynomial== | ||
| 40번째 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> |
| − | + | ||
==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 |
| 58번째 줄: | 58번째 줄: | ||
* [[topological quantum field theory(TQFT)]] | * [[topological quantum field theory(TQFT)]] | ||
| − | + | ||
| − | + | ||
==knot and QFT== | ==knot and QFT== | ||
| 71번째 줄: | 71번째 줄: | ||
* [[volume of hyperbolic threefolds and L-values]] | * [[volume of hyperbolic threefolds and L-values]] | ||
| − | + | ||
==computational resource== | ==computational resource== | ||
* https://docs.google.com/file/d/0B8XXo8Tve1cxUlVqT190VzRTdGs/edit | * https://docs.google.com/file/d/0B8XXo8Tve1cxUlVqT190VzRTdGs/edit | ||
| 80번째 줄: | 80번째 줄: | ||
* Atiyah, Michael The Geometry and Physics of Knots | * Atiyah, Michael The Geometry and Physics of Knots | ||
| − | + | ||
==encyclopedia== | ==encyclopedia== | ||
| 93번째 줄: | 93번째 줄: | ||
* [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 | ||
| 107번째 줄: | 107번째 줄: | ||
** Richard Altendorfer | ** Richard Altendorfer | ||
| − | + | ||
==question and answers(Math Overflow)== | ==question and answers(Math Overflow)== | ||
| 115번째 줄: | 115번째 줄: | ||
| − | + | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:Knot theory]] | [[분류:Knot theory]] | ||
[[분류:migrate]] | [[분류:migrate]] | ||
2020년 12월 28일 (월) 05:10 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
- 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