The Relation To Link(knot) Polynomials
At International Congress of Mathematicians in 2006 Mikhail Khovanov provided the following explanation for for the relation to knot polynomials from the view point of Khovanov homology. The skein relation for three links and is described as
Substituting leads to a link polynomial invariant, normalized so that for
and . For the polynomial can be interpreted via the representation theory of quantum group and via that of the quantum Lie superalgebra .
- The Alexander polynomial is the Euler characteristic of a bigraded knot homology theory.
- is trivial.
- The Jones polynomial is is the Euler characteristic of a bigraded link homology theory.
- The entire HOMFLY-PT polynomial is the Euler characteristic of a triply-graded link homology theory.
Read more about this topic: Khovanov Homology
Famous quotes containing the word relation:
“There is undoubtedly something religious about it: everyone believes that they are special, that they are chosen, that they have a special relation with fate. Here is the test: you turn over card after card to see in which way that is true. If you can defy the odds, you may be saved. And when you are cleaned out, the last penny gone, you are enlightened at last, free perhaps, exhilarated like an ascetic by the falling away of the material world.”
—Andrei Codrescu (b. 1947)