"Knot theory"의 두 판 사이의 차이

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45번째 줄: 45번째 줄:
  
 
* Kauffman bracket
 
* Kauffman bracket
 +
* colored Jones polynomial
 
* [[Hecke algebra]]
 
* [[Hecke algebra]]
 +
* [[Jones polynomials]] and <math>U_q[\mathfrak{sl}(2)]</math>
  
 
 
 
 
53번째 줄: 55번째 줄:
 
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Knot theory, statistical mechanics and quantum groups</h5>
 
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Knot theory, statistical mechanics and quantum groups</h5>
  
*  Jones polynomial and <math>U_q[\mathfrak{sl}(2)]</math><br>
+
* [[Knot theory|Knot Theory]] and Statistical Mechanics<br>
* [[Knot theory|Knot Theory]] and Statistical Mechanics[http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf ]<br>
 
 
** http://web.phys.ntu.edu.tw/phystalks/Wu.pdf
 
** http://web.phys.ntu.edu.tw/phystalks/Wu.pdf
  
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<h5>2+1 dimensional TQFT</h5>
 
<h5>2+1 dimensional TQFT</h5>
  
* [[topological quantum field theory(TQFT)|3D TQFT( Chern-Simons theory)]]
+
* [[topological quantum field theory(TQFT)]]
 
 
 
 
 
 
 
 
 
 
<h5>mathematica</h5>
 
 
 
# KnotData["Trefoil"]<br> KnotData["Trefoil", "JonesPolynomial"][x]
 
  
 
 
 
 

2010년 8월 11일 (수) 08:36 판

introduction

_2010_01_29_10136.jpg

 

Given a knot and a rational number one can define a closed three-manifold by Dehn surgery

 

 

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

 

 

history

 

 

related items

 

books

 

 

encyclopedia[1]

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

articles

 

 

experts on the field

 

 

TeX