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

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 \(U_q[\mathfrak{sl}(2)]\)

Knot theory, statistical mechanics and quantum groups

• Knot Theory and Statistical Mechanics
  • 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

2+1 dimensional TQFT

• topological quantum field theory(TQFT)

knot and QFT

• knot and quantum field theory

related items

• Knot theory and q-series
• volume of hyperbolic threefolds and L-values

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxUlVqT190VzRTdGs/edit

books

• Atiyah, Michael The Geometry and Physics of Knots

encyclopedia

• 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

articles

• A link invariant from quantum dilogarithm
  • Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418
• Knot theory and statistical mechanics
  • Richard Altendorfer
• http://www.bkfc.net/altendor/KnotTheoryAndStatisticalMechanics.pdf
• Knot and physics
  • Kauffman, 1989

• On knot invariants related to some statistical mechanical models.
  • V. F. R. Jones, 1989
• The Yang-Baxter equation and invariants of links
  • Turaev, 1988
• An Introduction to Knot Theory
  • Richard Altendorfer

question and answers(Math Overflow)

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

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