# Knot theory

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## 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

## 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

- 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

- 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

## computational resource

## 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)

## 메타데이터

### 위키데이터

- ID : Q849798

### Spacy 패턴 목록

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