Khovanov Homology

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The Khovanov Homology [math]\displaystyle{ \operatorname{\it KH}(L) }[/math] of a knot or a link [math]\displaystyle{ L }[/math], also known as Khovanov's categorification of the Jones polynomial of [math]\displaystyle{ L }[/math], was defined by Khovanov in [Khovanov1] (also check [Bar-Natan1]), where the notation is closer to the notation used here). It is a graded homology theory; each homology group [math]\displaystyle{ \operatorname{\it KH}^r(L) }[/math] is in itself a direct sum [math]\displaystyle{ \bigoplus_j\operatorname{\it KH}^r_j(L) }[/math] of homogeneous components. Over a field on can form the two-variable "Poincar'e polynomial" [math]\displaystyle{ \operatorname{\it Kh}(L) }[/math] (which deserves the name "the Khovanov polynomial of L"),

[math]\displaystyle{ \operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L) }[/math].

(For In[1] see Setup)

In[2]:= ?Kh Kh[L][q, t] returns the Poincare polynomial of the Khovanov Homology of a knot/link L (over a field of characteristic 0) in terms of the variables q and t. Kh[L, Program -> prog] uses the program prog to perform the computation. The currently available programs are "FastKh", written in Mathematica by Dror Bar-Natan in the winter of 2005 and "JavaKh" (default), written in java (java 1.5 required!) by Jeremy Green in the summer of 2005. The java program is several thousand times faster than the Mathematica program, though java may not be available on some systems. "JavaKh" also takes the option "Modulus -> p" which changes the characteristic of the ground field to p. If p==0 JavaKh works over the rational numbers; if p==Null JavaKh works over Z (see ?ZMod for the output format).

Thus for example, here's the Khovanov polynomial of the knot 5_1:

{{InOut| n = 3 | in = kh = Kh[Knot[5, 1]][q, t] | out= -5 -3 1 1 1 1 q + q