Knot theory

Pythagoras0 (토론 | 기여)님의 2021년 2월 17일 (수) 02:57 판
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  • 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


  • Atiyah, Michael The Geometry and Physics of Knots



question and answers(Math Overflow)



Spacy 패턴 목록

  • [{'LOWER': 'knot'}, {'LEMMA': 'theory'}]