Knot theory

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

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.

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