The Khovanov invariant is one of the hottest new knot invariants around. It is not a numerical invariant like crossing number or a polynomial-valued invariant like the Alexander polynomial but rather arises as the bigraded homology groups of a certain chain complex associated with any knot diagram. My presentation here is simplified in several ways. For example, I work with fields instead of rings (hence vector spaces instead of modules). For a more in-depth look at the invariant, look Khovanov up on the preprint archive (or Bar-Natan, whose summary is more readable for those of us who don't mind a little hand-waving).
To Khovanov invariant of a knot or diagram (henceforth referred to as Kh(K) or Kh(D) is often called a categorification of the Jones polynomial, but it would be more fitting to call it a categorification of the Kauffman Bracket <K> because computation of Kh(D) occurs in a manner parallel to that of the state sum model of <D>
We begin by generating the smoothings of the diagram. Locally, any crossing in a knot can be smoothed to change the diagram into a diagram with one fewer crossings in the following two ways:
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Crossing 0-Smoothing 1-Smoothing
In a diagram with k crossings, there are 2k possible total smoothings, each of which consists of some number N of disjoint circles. To each such total smoothing we give a height r equal to the number of 1-smoothings in it.
Now we are ready to start creating the groups for our chain complex. Let A be a graded vector space with two basis elements e+ and e-, whose degrees are +1 and -1. Then for each total smoothing, associate the tensor product of N copies of A (a 2N dimensional graded vector space whose elements vary in degree between -N and N inclusive and share N's parity). Direct sum those spaces which share the same height and raise the degree of each element by r, so that there are now k + 1 spaces, each of which is the direct sum of tensor products of A. To make this a chain complex, all that is neccesary is a differential d (This, of course, should be a delta, but my html skills need work)that takes the height h space into the height h+1 space and satisfies dd=0
Whenever two total smoothings differ by one smoothing, d carries the 0-smoothing space into the 1-smoothing space. This kind of one-smoothing difference always corresponds to a change by 1 in the number of disjoint circles, so every such relation will always be one of the following two types:
1. one of N disjoint circles splits into two circles, the others are unchanged
2. two of N disjoint circles join into one circle, the others are unchanged.
In either case, the tensor factors of the unchanged circles are unchanged by d, while the absolute action of d on the changed circles are:
1.
Del(e+) = (e+)x(e-) + (e-)x(e+)
Del(e-) = (e-)x(e-)
2.
m(e+)x(e+) = e+
m(e+)x(e-) = m(e-)x(e+) = e-
m(e-)x(e-) = 0
With the height shift as above and with the caveat that the differentials should be degree 0, these are determined up to constants. m and Del as written commute around any four smoothings related as follows:
S2
/ \
S1 S4
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S3
In order to make dd = 0, all that is necessary is that this diagram anti-commute, which is easily done by making one or three of the maps negative for each such "square." There are trivial ways of doing this which I will omit because they require complicated notation.
If a slight change in degree and height of the entire complex are made to compensate for the writhe of the original diagram, the homology groups of degree j and height i are now knot invariants up to isomorphism. What is more, the graded Euler characteristic Sigma((-1)iqj) is equal on change of variable and normalization to the Jones polynomial.
Conjecture, results, and significance of the Khovanov invariant