Lie algebras and the Four Color Theorem

@article{BarNatan1997LieAA,
  title={Lie algebras and the Four Color Theorem},
  author={Dror Bar-Natan},
  journal={Combinatorica},
  year={1997},
  volume={17},
  pages={43-52}
}
We present a statement about Lie algebras that is equivalent to the Four Color Theorem. 
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References

SHOWING 1-10 OF 28 REFERENCES
Map coloring and the vector cross product
An algebraic approach to the planar coloring problem
AbstractWe point out a general relationship between the planar coloring problem withQ colors and the Temperley-Lieb algebra with parameter $$\sqrt Q $$ . This allows us to give a complete algebraic…
MAP COLORING, q-DEFORMED SPIN NETWORKS, AND TURAEV-VIRO INVARIANTS FOR 3-MANIFOLDS
This paper reports on map coloring, q-deformed spin networks, and Turaev-viro m variants for 3-manifolds. Also discussed are errors, correction, m variance principle and equations.
On finite type 3-manifold invariants III: Manifold weight systems
Weights of Feynman diagrams and the Vassiliev knot invariants
Given a representation of a Lie algebra and an ad-invariant bilinear form we show how to assign numerical weights to a certain collection of graphs. This assignment is then shown to satisfy certain…
Chern-Simons Perturbation Theory
We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the…
The Trivial Connection Contribution Invariant and Finite Type Invariants of Rational Homology Spheres to Witten's
We derive an analog of the Melvin-Morton bound on the power series expansion of the colored Jones polynomial of algebraically split links and boundary links. This allows us to produce a simple…
The trivial connection contribution to Witten’s invariant and finite type invariants of rational homology spheres
We derive an analog of the Melvin-Morton bound on the power series expansion of the colored Jones polynomial of algebraically split links and boundary links. This allows us to produce a simple…
Chern-Simons perturbation theory. II
We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the…
Perturbative Aspects of the Chern-Simons Topological Quantum Field Theory
We investigate the Feynman-diagram perturbative expansion of the Chem-Simons topological quantum field theory. After introducing the theory, we compute the one-loop expectation value for knots and…
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