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~\cite{Khovanov:Categorification} (also check my paper~\cite{Bar-Natan:Categorification}, where the notation is much closer to the notation used here). It is a graded homology theory; each homology group $\KH^r(L)$ is in itself a direct sum $\bigoplus_j\KH^r_j(L)$ of homogeneous components. Over a field on can form the two-variable ``Poincar\'e polynomial $\Kh(L)$ (which deserves the name ``the Khovanov polynomial of $L$), \[ \Kh(L)(q,t) := \sum_{r,j}t^rq^j\dim\KH^r_j(L). \]

\latexhtml{\small (for {\tt In[1]} see Section~\ref{sec:Setup}.)}{\htmlref{{\tt In[1]}}{sec:Setup}} %<* InOut[1] *>

<* HelpBox[Kh] *>

Thus for example, here's the Khovanov polynomial of the knot \hlink{../Knots/5.1.html}{$5_1$}:

<*InOut@"kh = Kh[Knot[5, 1]][q, t]"*> \vskip 6pt

\index{Euler characteristic} The Euler characteristic of the Khovanov Homology $\KH(L)$ is (up to normalization) the Jones polynomial $J(L)$ of $L$. Precisely, \[ \Kh(L)(q, -1) = \hatJ(L)(q) := (q+q^{-1})J(L)(q^2). \] Let us verify this in the case of \hlink{../Knots/5.1.html}{$5_1$}: <*InOut@"{kh /. t -> -1, Expand[(q+1/q)Jones[Knot[5, 1]][q^2]]}"*> \vskip 6pt

Khovanov's homology is a strictly stronger invariant than the Jones polynomial. Indeed, $J(5_1)=J(10_{132})$ though $\Kh(5_1)\neq\Kh(10_{132})$:

<*InOut@"{\n

 Jones[Knot[5, 1]] === Jones[Knot[10, 132]],\n
 Kh[Knot[5, 1]] === Kh[Knot[10, 132]]\n

}"*> \vskip 6pt

The algorithm presently used by {\tt KnotTheory`} is an efficient algorithm modeled on the Kauffman bracket algorithm of Section~\ref{subsubsec:HowJones}, as explained in~\cite{Bar-Natan:ImHappy} (which follows~\cite{Bar-Natan:Cobordism}). Currently, two implementations of this algorithm are available:

\begin{itemize}

\index{FastKh@{\tt FastKh}} \item {\tt FastKh}: My original implementation, written in Mathematica in the winter of 2005. This implementation can be explicitly invoked using the syntax {\tt Kh[L, Program -> "FastKh"][q, t]} or by changing the default behaviour of {\tt Kh} by evaluating {\tt SetOptions[Kh, Program -> "FastKh"]}.

\index{JavaKh@{\tt JavaKh}} \index{Green, Jeremy} \item {\tt JavaKh}: In the summer of 2005 Jeremy Green re-implemented the algorithm in java {\bf (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 {\tt Kh[L, Program -> "JavaKh"][q, t]}.

{\tt JavaKh} takes an additional option, {\tt 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 {\tt In} lines imply that the Khovanov homology of the torus knot \hlink{../TorusKnots/6.5.html}{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}