Knot theory

introduction

• ambient isotopy defines an equivalence class of knots
• three Reidemeister moves

 

 

examples

• trivial knot
• Hopf link
• figure 8 knot
• trefoil knot

 

 

knot diagram

• projection to two dimensional space

 

 

 

 

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
• Hecke algebra

 

 

Knot theory, statistical mechanics and quantum groups

• Jones polynomial and \(U_q[\mathfrak{sl}(2)]\)
  
• Knot Theory and Statistical Mechanics[1]
  
  • 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
• so the problem is to find a nice set of Boltzmann weights which give non-trivial invariants

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

 

books

• The Geometry and Physics of Knots
  
  • Atiyah, Michael
• 찾아볼 수학책
• http://gigapedia.info/1/atiyah
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

encyclopedia[2]

• 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
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=knot+quantum
• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

•  
  Unknotting knot theory
  
  • Julie Rehmeyer, ScienceNews, October 31st, 2008
    
• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=partition+function+knot+theory
  • http://blogsearch.google.com/blogsearch?q=매듭이론
  • http://blogsearch.google.com/blogsearch?q=knot+theory
  • http://blogsearch.google.com/blogsearch?q=

 

 

articles

• The Jones Polynomial
  
  • V.Jones, 2005-8
• http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf
• Knot and physics
  
  • Kauffman, 1989
    
• Quantum field theory and the Jones polynomial
  
  • Edward Witten, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399

• On knot invariants related to some statistical mechanical models.
  
  • V. F. R. Jones, 1989
• The Yang-Baxter equation and invariants of links
  
  • Turaev, 1988
• 논문정리
• http://www.ams.org/mathscinet/search/publications.html?pg4=ALLF&s4=
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/

 

 

experts on the field

• http://arxiv.org/