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 $L$"),

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

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

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:

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

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].

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.