Knot theory

introduction

• 매듭이론 (knot theory)

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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

 

 

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
• colored Jones polynomial
• Hecke algebra
• Jones polynomials and \(U_q[\mathfrak{sl}(2)]\)