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Room P3.10, Mathematics Building
Marco Mackaay, Universidade do Algarve
Khovanov's categorification of the Jones polynomial
Following Bar-Natan's down-to-earth approach, I will explain
Khovanov's construction which associates to a knot a certain
complex of graded vector spaces. If two knots are ambient isotopic
their complexes are homotopy equivalent (grading is preserved).
Therefore the cohomology groups of the knot complex are
knot-invariants. It turns out that the Jones polynomial of a knot
equals the graded Euler characteristic of the knot cohomology.
Khovanov derived a more general polynomial from his complexes which
is a more powerful knot invariant, as has been shown by explicit
computations. References:
- Dror Bar-Natan, "On Khovanov's categorification of the Jones polynomial", Algebraic and Geometric Topology 2 (2002) 337-370; math.QA/0201043.
- Mikhail Khovanov, "A functor-valued invariant of tangles", Algebr. Geom. Topol. 2 (2002) 665-741; math.QA/0103190.
- Mikhail Khovanov, "A categorification of the Jones polynomial", math.QA/9908171.