Khovanov Homology

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The Khovanov Homology $\operatorname {\it {KH}} (L)$ of a knot or a link $L$, also known as Khovanov's categorification of the Jones polynomial of $L$, 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 $\operatorname {\it {KH}} ^{r}(L)$ is in itself a direct sum $\bigoplus _{j}\operatorname {\it {KH}} _{j}^{r}(L)$ of homogeneous components. Over a field on can form the two-variable "Poincar\'e polynomial" $\operatorname {\it {Kh}} (L)$ (which deserves the name "the Khovanov polynomial of $L$"),

$\operatorname {\it {Kh}} (L)(q,t):=\sum _{r,j}t^{r}q^{j}\dim \operatorname {\it {KH}} _{j}^{r}(L)$.

(For In[1] see Setup)

In[1]:= ?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:

In[2]:=	kh = Kh[Knot[5, 1]][q, t]
Out[2]=	-5 -3 1 1 1 1 q + q + ------ + ------ + ------ + ----- 15 5 11 4 11 3 7 2 q t q t q t q t

The Euler characteristic of the Khovanov Homology $\operatorname {\it {KH}} (L)$ is (up to normalization) the Jones polynomial $J(L)$ of $L$. Precisely,

$\operatorname {\it {Kh}} (L)(q,-1)={\hat {J}}(L)(q):=(q+q^{-1})J(L)(q^{2})$. \]

Let us verify this in the case of 5_1:

In[3]:=	{kh /. t -> -1, Expand[(q+1/q)Jones[Knot[5, 1]][q^2]]}
Out[3]=	-15 -7 -5 -3 -15 -7 -5 -3 {-q + q + q + q , -q + q + q + q }

Khovanov's homology is a strictly stronger invariant than the Jones polynomial. Indeed, $J(5_{1})=J(10_{132})$ though $\operatorname {\it {Kh}} (5_{1})\neq \operatorname {\it {Kh}} (10_{132})$:

In[4]:=	{ Jones[Knot[5, 1]] === Jones[Knot[10, 132]], Kh[Knot[5, 1]] === Kh[Knot[10, 132]] }
Out[4]=	{True, False}

The algorithm presently used by KnotTheory` is an efficient algorithm modeled on the Kauffman bracket algorithm of The_Jones_Polynomial#How_is_the_Jones_polynomial_computed.3F, as explained in [Bar-Natan3] (which follows [Bar-Natan2]). Currently, two implementations of this algorithm are available:

• FastKh: My original implementation, written in Mathematica in the winter of 2005. This implementation can be explicitly invoked using the syntax Kh[L, Program -> "FastKh"][q, t] or by changing the default behaviour of Kh by evaluating SetOptions[Kh, Program -> "FastKh"].
• JavaKh: In the summer of 2005 Jeremy Green re-implemented the algorithm in java (java 1.5 required!) with much further care to the details, leading to an improvemnet factor of several thousands for large knots/links. This implementation is the default. It can also be explicitly invoked from within Mathematica using the syntax Kh[L, Program -> "JavaKh"][q, t].

In[5]:=	Options[Kh]
Out[5]=	{ExpansionOrder -> Automatic, Program -> JavaKh, Modulus -> 0, JavaOptions -> }

JavaKh takes an additional option, Modulus, which sets the characteristic of the ground field for the homology computations to $0$ or to a prime $p$. Thus for example, the following four In lines imply that the Khovanov homology of the torus knot T(6,5) has both 3 torsion and 5 torsion, but no 7 torsion:

<*InOut@"T65 = TorusKnot[6, 5]; kh = Kh[T65][q, t];" *> % <* (* Cheat: *)

 Kh[TorusKnot[6, 5], Modulus->3] = Function @@ {
   kh + q^43*t^13 + q^43*t^14 /. {q->#1, t->#2}
 };
 Kh[TorusKnot[6, 5], Modulus->5] = Function @@ {
   kh + q^35*t^10 + q^35*t^11 + q^39*t^11 + q^39*t^12 /. {q->#1, t->#2}
 };
 Kh[TorusKnot[6, 5], Modulus->7] = Function @@ {kh /. {q->#1, t->#2}};

• >

<*InOut@"Kh[T65, Modulus -> 3][q, t] - kh"*> <*InOut@"Kh[T65, Modulus -> 5][q, t] - kh"*> <*InOut@"Kh[T65, Modulus -> 7][q, t] - kh"*> \vskip 6pt

The following further example is a bit tougher. It takes my computer nearly an hour and some 256Mb of memory to find that the Khovanov homology of the 48-crossing torus knot T(8,7) has 3, 5 and 7 torsion but no 11 torsion:

<* HelpBox[JavaOptions] *>

<*InOut@"SetOptions[Kh, JavaOptions -> \"-Xmx256m\"];" *> % <* (* Cheat: *)

 Kh[TorusKnot[8, 7]] = Function @@ {
   (
     q^41 + q^43 + q^45*t^2 + q^49*t^3 + q^47*t^4 + q^49*t^4 + q^51*t^5 +
     q^53*t^5 + q^49*t^6 + q^51*t^6 + q^53*t^7 + q^55*t^7 + q^51*t^8 +
     2*q^53*t^8 + q^55*t^9 + 2*q^57*t^9 + q^53*t^10 + 2*q^55*t^10 +
     q^57*t^11 + 3*q^59*t^11 + q^55*t^12 + 3*q^57*t^12 + q^59*t^12 +
     q^63*t^12 + q^59*t^13 + 4*q^61*t^13 + q^63*t^13 + 2*q^59*t^14 +
     q^61*t^14 + q^65*t^14 + 4*q^63*t^15 + 2*q^65*t^15 + 2*q^61*t^16 +
     2*q^63*t^16 + 2*q^67*t^16 + q^69*t^16 + 3*q^65*t^17 + 3*q^67*t^17 +
     q^63*t^18 + 2*q^65*t^18 + q^69*t^18 + q^71*t^18 + 2*q^67*t^19 +
     3*q^69*t^19 + q^65*t^20 + 2*q^67*t^20 + q^71*t^20 + q^73*t^20 +
     q^69*t^21 + 3*q^71*t^21 + q^69*t^22 + q^75*t^22 + 2*q^73*t^23 +
     q^71*t^24 + q^73*t^24 + q^77*t^24 + q^75*t^25 + q^77*t^25
   ) /. {q->#1, t->#2}
 };

• >

<*InOut@"T87 = TorusKnot[8, 7]; kh = Kh[T87][q, t];" *> % <* (* Cheat: *)

 Kh[TorusKnot[8, 7], Modulus->3] = Function @@ {
   kh + q^79*t^25 + q^79*t^26 /. {q->#1, t->#2}
 };
 Kh[TorusKnot[8, 7], Modulus->5] = Function @@ {
   kh + (
     q^61*t^11 + q^61*t^12 + q^73*t^21 + q^73*t^22 + q^75*t^23 +
     q^75*t^24 + q^79*t^24 + q^79*t^25
   ) /. {q->#1, t->#2}
 };
 Kh[TorusKnot[8, 7], Modulus->7] = Function @@ {
   kh + (
     q^61*t^14 + q^61*t^15 + q^69*t^20 + q^69*t^21 + q^73*t^21 +
     q^71*t^22 + q^73*t^22 + q^71*t^23 + q^75*t^23 + q^75*t^24
   ) /. {q->#1, t->#2}
 };
 Kh[TorusKnot[8, 7], Modulus->11] = Function @@ {kh /. {q->#1, t->#2}};

• >

<*InOut@"Factor[Kh[T87, Modulus -> 3][q, t] - kh]"*> <*InOut@"Factor[Kh[T87, Modulus -> 5][q, t] - kh]"*> <*InOut@"Factor[Kh[T87, Modulus -> 7][q, t] - kh]"*> <*InOut@"Factor[Kh[T87, Modulus -> 11][q, t] - kh]"*>

{\tt JavaKh} also works over the integers:

<* HelpBox[ZMod] *>

For example, the 22nd homology group over $\bbZ$ of the torus knot T(8,7) at degree 73 is the 280 element torsion group $\bbZ_2\oplus\bbZ_4\oplus\bbZ_5\oplus\bbZ_7$: % <* (* Cheat: *)

 Kh[TorusKnot[8, 7], Modulus->Null] = Function @@ {
   q^73*t^22*ZMod[2, 4, 5, 7] /. {q->#1, t->#2}
 };

• >

<*InOut@"Coefficient[Kh[T87, Modulus -> Null][q, t], t^22 * q^73]"*> \vskip 6pt

Finally, {\tt JavaKh} may also be run outside of Mathematica, as the following example demonstrates: \begin{verbatim} drorbn@coxeter:.../KnotTheory: cd JavaKh drorbn@coxeter:.../KnotTheory/JavaKh: java JavaKh PD[X[3, 1, 4, 6], X[1, 5, 2, 4], X[5, 3, 6, 2]] "+ q^1t^0 + q^3t^0 + q^5t^2 + q^9t^3 " \end{verbatim}

\noindent (Type {\tt java JavaKh -help} for some further help).

\end{itemize}

\begin{figure} \begin{center} \latex{

 \includegraphics[width=3in]{figs/MikhailKhovanov.ps}

} \begin{rawhtml}

 <img src=MikhailKhovanov.jpg alt="Mikhail Khovanov">

\end{rawhtml} \end{center} \caption{

 August 2002, Toronto: Mikhail Khovanov explaining his more recent
 paper~\cite{Khovanov:Cobordisms}.

} \label{fig:MikhailKhovanov} \end{figure}

[Bar-Natan1] ^  D. Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebraic and Geometric Topology 2-16 (2002) 337-370, arXiv:math.GT/0201043.

[Bar-Natan2] ^  D. Bar-Natan, Khovanov's Homology for Tangles and Cobordisms, Geometry and Topology 9-33 (2005) 1443-1499, arXiv:math.GT/0410495.

[Bar-Natan3] ^  D. Bar-Natan, I've Computed Kh(T(9,5)) and I'm Happy, talk given at Knots in Washington XX, George Washington University, February 2005.

[Khovanov1] ^  M. Khovanov, A categorification of the Jones polynomial, arXiv:math.QA/9908171.