# Survey of Finite Type Invariants

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\begin{document}

\title{Finite Type Invariants}

\author{Dror~Bar-Natan} \address{

Department of Mathematics\\ University of Toronto\\ Toronto Ontario M5S 3G3\\ Canada

} \email{drorbn@math.toronto.edu} \urladdr{http://www.math.toronto.edu/\~{}drorbn}

\date{

First edition: Aug.~13,~2004. This edition: \today

}

\thanks{This work was partially supported by NSERC grant RGPIN 262178.

Electronic versions: {\tt http://\linebreak[0]www.math.toronto.edu/\linebreak[0]$\sim$drorbn/\linebreak[

0]papers/\linebreak[0]EMP/} and arXiv:math.GT/0408182.

Copyright is retained by the author.

}

\ifemp{}{

\begin{abstract} This is an overview article on finite type invariants, written for the {\em Encyclopedia of Mathematical Physics}. \end{abstract}

}

\maketitle

\begin{multicols}{2}

\section{Introduction} \label{sec:intro}

Knots belong to sailors and climbers and upon further reflection,
perhaps also to geometers, topologists or combinatorialists.
Surprisingly, throughout the 1980s it became apparent that knots are
also closely related to several other branches of mathematics in
general and mathematical physics in particular. Many of these
connections (though not all!) factor through the notion of ``finite
type invariants* (aka ``Vassiliev* or ``Goussarov-Vassiliev
invariants)~\Cite{Goussarov~91,~93, Vassiliev~90,~92, Birman-Lin~93,
Kontsevich~93, Bar-Natan~95}{Goussarov:New, Goussarov:nEquivalence,
Vassiliev:CohKnot, Vassiliev:Book, BirmanLin:Vassiliev,
Kontsevich:Vassiliev, Bar-Natan:OnVassiliev}.

Let $V$ be an arbitrary invariant of oriented knots in oriented space with values in some Abelian group $A$. Extend $V$ to be an invariant of $1$-singular knots, knots that may have a single singularity that locally looks like a double point \lrg{\doublepoint}, using the formula \begin{equation} \label{eq:doublepoint}

V(\lrg{\doublepoint})=V(\lrg{\overcrossing})-V(\lrg{\undercrossing}).

\end{equation} Further extend $V$ to the set $\calK^m$ of $m$-singular knots (knots with $m$ double points) by repeatedly using~\eqref{eq:doublepoint}.

\begin{definition} We say that $V$ is of type $m$ if its extension $\left.V\right|_{\calK^{m+1}}$ to $(m+1)$-singular knots vanishes identically. We say that $V$ is of finite type if it is of type $m$ for some $m$. \end{definition}

Repeated differences are similar to repeated derivatives and hence it
is fair to think of the definition of $\left.V\right|_{\calK^m}$ as
repeated differentiation. With this in mind, the above definition
imitates the definition of polynomials of degree $m$. Hence finite type
invariants can be thought of as ``polynomials* on the space of knots.*

As we shall see below, finite type invariants are plenty and powerful
and they carry a rich algebraic structure and are deeply related to Lie
algebras (Section~\ref{sec:Basic}). There are several constructions for
a ``universal finite type invariant* and those are related to*
conformal field theory, the Chern-Simons-Witten topological quantum
field theory and Drinfel'd's theory of associators and quasi-Hopf
algebras (Section~\ref{sec:Fundamental}). Finite type invariants have
been studied extensively (Section~\ref{sec:FurtherDirections}) and
generalized in several directions (Section~\ref{sec:BeyondKnots}). But
the first question on finite type invariants remains unanswered:

\begin{problem} Honest polynomials are dense in the space of functions. Are finite type invariants dense within the space of all knot invariants? Do they separate knots? \end{problem}

In a similar way one may define finite type invariants of framed knots (and ask the same questions).

\subsection{Acknowledgement} I wish to thank O.~Dasbach and A.~Savage for comments and corrections.

\section{Basic facts} \label{sec:Basic}

\subsection{Classical knot polynomials} The first (non trivial!) thing to
notice is that there are plenty of finite type invariants and they are at
least as powerful as all the standard knot polynomials\footnote{Finite type
invariants are like polynomials on the space of knots; the standard phrase
``knot polynomials* refers to a different thing --- knot invariants with*
polynomial values.} combined:

\begin{theorem} \Cite{Bar-Natan~95, Birman-Lin~93}{Bar-Natan:OnVassiliev, BirmanLin:Vassiliev} Let $J(K)(q)$ be the Jones polynomial of a knot $K$ (it is a Laurent polynomial in a variable $q$). Consider the power series expansion $J(K)(e^x)=\sum_{m=0}^\infty V_m(K)x^m$. Then each coefficient $V_m(K)$ is a finite type knot invariant. (And thus the Jones polynomial can be reconstructed from finite type information). \end{theorem}

A similar theorem holds for the Alexander-Conway, HOMFLY-PT and Kauffman polynomials~\Cite{Bar-Natan~95}{Bar-Natan:OnVassiliev}, and indeed, for arbitrary Reshetikhin-Turaev invariants~\cite{ReshetikhinTuraev:Ribbon}, \Cite{Lin~91}{Lin:QuantumGroups}. Though it is still unknown if the signature of a knot can be expressed in terms of its finite type invariants.

\subsection{Chord diagrams and the Fundamental Theorem} \label{subsec:ChordDiagrams} The top derivatives of a multi-variable polynomial form a system of constants that determine that polynomial up to polynomials of lower degree. Likewise the $m$th derivative $V^{(m)}:=V\left(\lrg{\doublepoint}\overset{m}{\cdots}\lrg{\doublepoint}\right)$

of a type $m$

invariant $V$ is a constant (for
$V\left(\lrg{\doublepoint}\overset{m}{\cdots}\lrg{\doublepoint}\lrg{\overcrossin
g}\right)
-V\left(\lrg{\doublepoint}\overset{m}{\cdots}\lrg{\doublepoint}\lrg{\undercrossi
ng}\right)
=V\left(\lrg{\doublepoint}\overset{m+1}{\cdots}\lrg{\doublepoint}\right)=0$ so $
V^{(m)}$ is
blind to 3D topology) and likewise $V^{(m)}$ determines $V$ up to
invariants of lower type. Hence a primary tool in the study of finite
type invariants is the study of the ``top derivative* $V^{(m)}$, also*
known as ``the weight system of $V$*.*

\parpic[r]{$\eps{width=4cm}{ToChordDiagrams}$}
Blind to 3D topology, $V^{(m)}$ only sees the combinatorics of the
circle that parametrizes an $m$-singular knot. On this circle there are $m$
pairs of points that are pairwise identified in the image; standardly one
indicates those by drawing a circle with $m$ chords marked (an ``$m$-chord
diagram*) as on the right.*

\begin{definition} Let $\calD_m$ denote the space of all formal linear combinations with rational coefficients of $m$-chord diagrams. Let $\calA^r_m$ be the quotient of $\calD_m$ by all $4T$ and $FI$ relations as drawn below (full details in e.g.~\Cite{Bar-Natan~95}{Bar-Natan:OnVassiliev}), and let $\hat\calA^r$ be the graded completion of $\calA:=\bigoplus_m\calA^r_m$. Let $\calA_m$, $\calA$ and $\hat\calA$ be the same as $\calA^r_m$, $\calA^r$ and $\hat\calA^r$ but without imposing the $FI$ relations. \[ 4T:\ \eps{height=8mm}{4T}\quad FI:\ \eps{height=8mm}{FI} \] \end{definition}

\begin{theorem} \label{thm:Fundamental} (The Fundamental Theorem)

\noindent $\bullet$\ (Easy part, \Cite{Vassiliev~90, Goussarov~91, Birman-Lin~93}{Vassiliev:CohKnot, Goussarov:New, BirmanLin:Vassiliev}). If $V$ is a rational valued type $m$ invariant then $V^{(m)}$ defines a linear functional on $\calA^r_m$. If in addition $V^{(m)}\equiv 0$, then $V$ is of type $m-1$.

\noindent $\bullet$\ (Hard part, \Cite{Kontsevich~93}{Kontsevich:Vassiliev} and Section~\ref{sec:Fundamental}). For any linear functional $W$ on $\calA^r_m$ there is a rational valued type $m$ invariant $V$ so that $V^{(m)}=W$. \end{theorem}

Thus to a large extent the study of finite type invariants is reduced to the finite (though super exponential in $m$) algebraic study of $\calA^r_m$. A similar theorem reduces the study of finite type invariants of framed knots to the study of $\calA_m$.

\subsection{The structure of $\calA$} Knots can be multiplied (the
``connected sum* operation) and knot invariants can be multiplied.*
This structure interacts well with finite type invariants and induces
the following structure on $\calA^r$ and $\calA$:

\begin{theorem} \Cite{Kontsevich~93, Bar-Natan~95, Willerton~96, Chmutov-Duzhin-Lando~94}{Kontsevich:Vassiliev, Bar-Natan:OnVassiliev, Willerton:Hopf, ChmutovDuzhinLando:VasI} $\calA^r$ and $\calA$ are commutative and cocommutative graded bialgebras (i.e., each carries a commutative product and a compatible cocommutative coproduct). Thus both $\calA^r$ and $\calA$ are graded polynomial algebras over their spaces of primitives, $\calP^r=\oplus_m\calP^r_m$ and $\calP=\oplus_m\calP_m$. \end{theorem}

Framed knots differ from knots only by a single integer parameter (the
``self linking*, itself a type $1$ invariant). Thus $\calP^r$ and $\calP$*
are also closely related.

\begin{theorem} \Cite{Bar-Natan~95}{Bar-Natan:OnVassiliev} $\calP=\calP^r\oplus\langle\theta\rangle$, where $\theta$ is the unique $1$-chord diagram $\eps{width=5mm}{theta}$. \end{theorem}

\subsection{Bounds and computational results} The following table (taken from~\Cite{Bar-Natan~95, Kneissler~97}{Bar-Natan:OnVassiliev, Kneissler:Twelve}) shows the number of type $m$ invariants of knots and framed knots modulo type $m-1$ invariants ($\dim\calA_m^r$ and $\dim\calA_m$) and the number of multiplicative generators of the algebra $\calA$ in degree $m$ ($\dim\calP_m$) for $m\leq 12$. Some further tabulated results are in~\Cite{Bar-Natan~96}{Bar-Natan:Computations}.

\begin{center} {\small

\def\n#1Template:$\!\! \begin{tabular}{||c|c|c|c|c|c|c|c|c|c|c|c|c|c||} \hline \n{m} & \n{0} & \n{1} & \n{2} & \n{3} & \n{4} & \n{5} & \n{6} & \n{7} & \n{8} & \n{9} & \n{10} & \n{11} & \n{12} \\ \hline \n{\dim\calA_m^r} & \n{1} & \n{0} & \n{1} & \n{1} & \n{3} & \n{4} & \n{9} & \n{14} & \n{27} & \n{44} & \n{80} & \n{132} & \n{232} \\ \n{\dim\calA_m} & \n{1} & \n{1} & \n{2} & \n{3} & \n{6} & \n{10} & \n{19} & \n{33} & \n{60} & \n{104} & \n{184} & \n{316} & \n{548} \\ \n{\dim\calP_m} & \n{0} & \n{1} & \n{1} & \n{1} & \n{2} & \n{3} & \n{5} & \n{8} & \n{12} & \n{18} & \n{27} & \n{39} & \n{55} \\ \hline \end{tabular}

} \end{center}

Little is known about these dimensions for large $m$. There is an explicit conjecture in~\Cite{Broadhurst~97}{Broadhurst:ConjecturedEnumeration} but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:

\begin{theorem} For large $m$, $\dim\calP_m>e^{c\sqrt m}$ (for any fixed $c<\pi\sqrt{\frac23}$) \Cite{Dasbach~00, Kontsevich (unpublished)}{Dasbach:CombinatorialStructureIII, Kontsevich:Unpublished} and $\dim\calA_m<6^mm!\sqrt{m}/\pi^{2m}$ \Cite{Stoimenow~98, Zagier~01}{Stoimenow:Enumeration, Zagier:StrangeIdentity}. \end{theorem}

\subsection{Jacobi diagrams and the relation with Lie algebras} \label{subsec:Jacobi} Much of the richness of finite type invariants stems from their relationship with Lie algebras. Theorem~\ref{thm:Jacobi} below suggests this relationship on an abstract level, Theorem~\ref{thm:Lie} makes that relationship concrete and Theorem~\ref{thm:PBW} makes it a bit deeper.

\begin{theorem} \label{thm:Jacobi}
\Cite{Bar-Natan~95}{Bar-Natan:OnVassiliev} The algebra $\calA$ is
isomorphic to the algebra $\calA^t$ generated by ``Jacobi diagrams in a
circle* (chord diagrams that are also allowed to have oriented*
internal trivalent vertices) modulo the $AS$, $STU$ and $IHX$
relations. See the figure below.
\end{theorem}

\vskip -1mm \[ \eps{width=2.8in}{Jacobi} \] \vskip 1mm

Thinking of trivalent vertices as graphical analogs of the Lie bracket, the $AS$ relation become the anti-commutativity of the bracket, $STU$ become the equation $[x,y]=xy-yx$ and $IHX$ becomes the Jacobi identity. This analogy is made concrete within the proof of the following:

\begin{theorem} \label{thm:Lie} \Cite{Bar-Natan~95}{Bar-Natan:OnVassiliev} Given a finite dimensional metrized Lie algebra $\frakg$ (e.g., any semi-simple Lie algebra) there is a map $\calT_\frakg:\calA\to\calU(\frakg)^\frakg$ defined on $\calA$ and taking values in the invariant part $\calU(\frakg)^\frakg$ of the universal enveloping algebra $\calU(\frakg)$ of $\frakg$. Given also a finite dimensional representation $R$ of $\frakg$ there is a linear functional $W_{\frakg,R}:\calA\to\bbQ$. \end{theorem}

The last assertion along with Theorem~\ref{thm:Fundamental} show that associated with any $\frakg$, $R$ and $m$ there is a weight system and hence a knot invariant. Thus knots are unexpectedly linked with Lie algebras.

The hope \Cite{Bar-Natan~95}{Bar-Natan:OnVassiliev} that all finite type invariants arise in this way was dashed by~\Cite{Vogel~97,~99, Lieberum~99}{Vogel:Structures, Vogel:UniversalLieAlgebra, Lieberum:NotComing}. But finite type invariants that do not arise in this way remain rare and not well understood.

The Poincar\'e-Birkhoff-Witt (PBW) theorem of the theory of Lie
algebras says that the obvious ``symmetrization* map*
$\chi_\frakg:\calS(\frakg)\to\calU(\frakg)$ from the symmetric algebra
$\calS(\frakg)$ of a Lie algebra $\frakg$ to its universal enveloping
algebra $\calU(\frakg)$ is a $\frakg$-module isomorphism. The following
definition and theorem form a diagrammatic counterpart of this theorem:

\parpic(12mm,13mm)[r]{\raisebox{-14mm}{$

\eps{width=11mm}{BExample}

$}}
\begin{definition}
Let $\calB$ be the space of formal linear
combinations of ``free Jacobi diagrams* (Jacobi diagrams as before,*
but with unmarked univalent ends (``legs*) replacing the circle; see*
an example on the right), modulo the $AS$ and $IHX$ relations of
before. Let $\chi:\calB\to\calA$ be the symmetrization map which maps a
$k$-legged free Jacobi diagram to the average of the $k!$ ways of
planting these legs along a circle.
\end{definition}

\parpic(28mm,20mm)[r]{\raisebox{-2mm}{$\xymatrix{

\calB \ar[r]^\chi \ar[d]^{\calT_\frakg} & \calA \ar[d]^{\calT_\frakg} \\ \calS(\frakg) \ar[r]^{\chi_\frakg} & \calU(\frakg)

}$}} \begin{theorem} \label{thm:PBW} (diagrammatic PBW, \Cite{Kontsevich~93, Bar-Natan~95}{Kontsevich:Vassiliev, Bar-Natan:OnVassiliev}) $\chi$ is an isomorphism of vector spaces. Furthermore, fixing a metrized $\frakg$ there is a commutative square as on the right. \end{theorem}

Note that $\calB$ can be graded (by half the number of vertices in a Jacobi diagram) and that $\chi$ respects degrees so it extends to an isomorphism $\chi:\hat\calB\to\hat\calA$ of graded completions.

\section{Proofs of the Fundamental Theorem} \label{sec:Fundamental}

The heart of all known proofs of Theorem~\ref{thm:Fundamental} is always a
construction of a ``universal finite type invariant* (see below); it is*
simple to show that the existence of a universal finite type invariant is
equivalent to Theorem~\ref{thm:Fundamental}.

\begin{definition} A universal finite type invariant is a map $Z:\{\text{knots}\}\to\hat\calA^r$ whose extension to singular knots satisfies $Z(K)=D+(\text{higher degrees})$ whenever a singular knot $K$ and a chord diagram $D$ are related as in Section~\ref{subsec:ChordDiagrams}. \end{definition}

\subsection{The Kontsevich Integral} \label{sec:KontsevichIntegral}
The first construction of a universal
finite type invariant was given by Kontsevich~\Cite{Kontsevich~93}{Kontsevich:Va
ssiliev}
(see also~\Cite{Bar-Natan~95, Chmutov-Duzhin~01}{Bar-Natan:OnVassiliev, ChmutovD
uzhin:KontsevichIntegral}).
It is known as ``the Kontsevich Integral* and up to a normalization factor*
it is given by\footnote{

The symbol $\sumint$ means ``sum over all discrete variables and integrate over all continuous variables.

} \[ Z_1(K)=\sum_{m=0}^\infty \frac{1}{(2\pi i)^m}

\hspace{-4mm} \sumint_{\stackrel{t_1<\ldots<t_m}{P=\{(z_i,z'_i)\}}} \hspace{-4mm} (-1)^{\#P_{\downarrow}}D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i},

\] where the relationship between the knot $K$, the pairing $P$, the real variables $t_i$, the complex variables $z_i$ and $z'_i$ and the chord diagram $D_P$ is summarized by the figure \[ \eps{width=3in}{DP}. \]

The Kontsevich Integral arises from studying the holonomy of the Knizhnik-Zamolodchikov equation of conformal field theory~\cite{KnizhnikZamolodchikov:CurrentAlgebra}. When evaluating $Z_1$ one encounters multiple $\zeta$-numbers~\Cite{Le-Murakami~95}{LeMurakami:HOMFLY} in a substantial way, and the proof that the end result is rational is quite involved~\Cite{Le-Murakami~96}{LeMurakami:Universal} and relies on deep results about associators and quasitriangular Quasi-Hopf algebras~\cite{Drinfeld:QuasiHopf, Drinfeld:GalQQ}. Employing the same techniques, in~\Cite{Le-Murakami~96}{LeMurakami:Universal} it is also shown that the composition of $W_{\frakg,R}\circ Z_1$ precisely reproduces the Reshetikhin Turaev invariants~\cite{ReshetikhinTuraev:Ribbon}.

\subsection{Perturbative Chern-Simons-Witten theory and configuration space integrals} \label{sec:CS} Historically the first approach to the construction of a universal finite type invariant was to use perturbation theory with the Chern-Simons-Witten topological quantum field theory; this is also how the relationship with Lie algebras first arose~\Cite{Bar-Natan~91}{Bar-Natan:Thesis}. But taming the integrals involved t urned out to be difficult and working constructions using this approach appeared only a bit later~\Cite{Bott-Taubes~94, Thurston~95, Altschuler-Freidel~97}{BottTaubes:SelfLinking, Thurston:IntegralExpressions, AltschulerFreidel:AllOrders}.

In short, one writes a perturbative expansion for the large $k$ asymptotics of the Chern-Simons-Witten path integral for some metrized Lie algebra $\frakg$ with a Wilson loop in some representation $R$ of $\frakg$, \[

\int_{\frakg\mbox{\scriptsize -connections}} \hspace{-42pt} \calD A\, \mbox{\it tr}_R\mbox{\it hol}_K(A)\exp\left[{\scriptstyle \frac{ik}{4\pi}\int\limits_{\bbR^3} \mbox{tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right) }\right].

\] The result is of the form \[

\sum_{\parbox{0.7in}{\centering\scriptsize $D$: Feynman diagram}} W_{\frakg(D),R}\sumint\calE(D)

\] where $\calE(D)$ is a very messy integral expression and the diagrams $D$ as well as the weights $W_{\frakg(D),R}$ are as in Section~\ref{subsec:Jacobi}. Replacing $W_{\frakg(D),R}$ by simply $D$ in the above formula we get an expression with values in $\hat\calA$: \[ Z_2(K):=\sum_D D\sumint\calE(D)\in\hat\calA. \]

For formal reasons $Z_2(K)$ ought to be a universal finite type invariant, and after much work taming the $\calE(D)$ factors and after multiplying by a further framing-dependent renormalization term $Z^{\text{anomaly}}$, the result is indeed a universal finite type invariant.

Upon further inspection the $\calE(D)$ factors can be reinterpreted as
integrals of certain spherical volume forms on certain (compactified)
configuration spaces~\Cite{Bott-Taubes~94}{BottTaubes:SelfLinking}.
These integrals can be further interpreted as counting certain ``tinker
toy constructions* built on top of*
$K$~\Cite{Thurston~95}{Thurston:IntegralExpressions}. The latter
viewpoint makes the construction of $Z_2$ visually
appealing~\Cite{Bar-Natan~00}{Bar-Natan:AstrologyToTopology}, but there
is no satisfactory writeup of this perspective yet.

We note that the precise form of the renormalization term $Z^{\text{anomaly}}$ remains an open problem. An appealing conjecture is that $Z^{\text{anomaly}}=\exp\frac12\isolatedchord$. If this is true then $Z_2=Z_1$ \Cite{Poirier~99}{Poirier:LimitConfigurationSpace}; but the conjecture is only verified up to degree 6 \Cite{Lescop~01}{Lescop:ForLinks} (there's also an unconfirmed verification to all orders~\Cite{Yang~97}{Yang:DegreeTheory}).

The most important open problem about perturbative Chern-Simons-Witten theory is not directly about finite type invariants, but it is nevertheless worthwhile to recall it here:

\begin{problem} Does the perturbative expansion of the Chern-Simons-Witten theory converge (or is asymptotic to) the exact solution due to Witten~\cite{Witten:Jones} and Reshetikhin-Turaev~\cite{ReshetikhinTuraev:Ribbon} when the parameter $k$ converges to infinity? \end{problem}

\subsection{Associators and trivalent graphs} There is also an entirely algebraic approach for the construction of a universal finite type invariant $Z_3$. The idea is to find some algebraic context within which knot theory is finitely presented --- i.e., presented by finitely many generators subject to finitely many relations. If the algebraic context at hand is compatible with the definitions of finite type invariants and of chord diagrams, one may hope to define $Z_3$ by defining it on the generators in such a way that the relations are satisfied. Thus the problem of defining $Z_3$ is reduced to finding finitely many elements of $\calA$-like spaces which solve certain finitely many equations.

A concrete realization of this idea is in~\Cite{Le-Murakami~96,
Bar-Natan~97}{LeMurakami:Universal,
Bar-Natan:NAT} (following ideas from~\cite{Drinfeld:QuasiHopf,
Drinfeld:GalQQ} on quasitriangular Quasi-Hopf algebras). The relevant
``algebraic context* is a category with*
certain extra operations, and within it, knot theory is generated by
just two elements, the braiding $\lrg{\overcrossing}$ and the re-association
$\lrg{\Associator}$. Thus to define $Z_3$ it is enough to find
$R=Z_3(\lrg{\overcrossing})$ and ``an associator* $\Phi=Z_3(\lrg{\Associator})$*

which

satisfy certain normalization conditions as well as the pentagon and hexagon equations \[

\Phi^{123}\cdot(1\Delta 1)(\Phi)\cdot\Phi^{234} =(\Delta 1 1)(\Phi)\cdot (1 1\Delta)(\Phi),

\] \[

(\Delta 1)(R^{\pm}) = \Phi^{123} (R^{\pm})^{23}(\Phi^{-1})^{132} (R^{\pm})^{13}\Phi^{312}.

\]

As it turns out the solution for $R$ is easy and nearly canonical. But finding an associator $\Phi$ is a lot harder. There is a closed form integral expression $\Phi^{KZ}$ due to~\cite{Drinfeld:QuasiHopf} but one encounters the same not-too-well-understood multiple $\zeta$ numbers of Section~\ref{sec:KontsevichIntegral}. There is a rather complicated iterative procedure for finding an associator~\cite{Drinfeld:GalQQ, Bar-Natan:Associators}, \Cite{Bar-Natan~97}{Bar-Natan:NAT}. On a computer it had been used to find an associator up to degree $7$. There is also closed form associator that works only with the Lie super-algebra $gl(1|1)$~\Cite{Lieberum~02}{Lieberum:gl11}. But it remains an open problem to find a closed form formula for a rational associator (existence by~\cite{Drinfeld:GalQQ, Bar-Natan:Associators}).

On the positive side we should note that the end result, the invariant $Z_3$, is independent of the choice of $\Phi$ and that $Z_3=Z_1$~\Cite{Le-Murakami~96}{LeMurakami:Universal}.

There is an alternative (more symmetric and intrinsicly 3-dimensional, but less well documented) description of the theory of associators in terms of knotted trivalent graphs~\cite{Bar-NatanThurston:AlgebraicStructures}, \Cite{Thurston~01}{Thurston:Shadow}. There ought to be a perturbative invariant associated with knotted trivalent graphs in the spirit of Section~\ref{sec:CS} and such an invariant should lead to a simple proof that $Z_2=Z_3=Z_1$. But the $\calE(D)$ factors remain untamed in this case.

\subsection{Step by step integration} \label{subsec:StepByStep} The last approach for proving the Fundamental Theorem is the most natural and historically the first. But here it is last because it is yet to lead to an actual proof. A weight system $W:\calA^r_m\to\bbQ$ is an invariant of $m$-singular knots. We want to show that it is the $m$th derivative of an invariant $V$ of non-singular knots. It is natural to try to integrate $W$ step by step, first finding an invariant $V^{m-1}$ of $(m-1)$-singular knots whose derivative in the sense of~\eqref{eq:doublepoint} is $W$, then an invariant $V^{m-2}$ of $(m-2)$-singular knots whose derivative is $V^{m-1}$, and so on all the way up to an invariant $V^0=V$ whose $m$th derivative will then be $W$. If proven, the following conjecture would imply that such an inductive procedure can be made to work:

\begin{conjecture} \Cite{Hutchings~98}{Hutchings:SingularBraids} If $V^r$ is a once-integrable invariant of $r$-singular knots then it is also twice integrable. That is, if there is an invariant $V^{r-1}$ of $(r-1)$-singular knots whose derivative is $V^r$, then there is an invariant $V^{r-2}$ of $(r-2)$-singular knots whose second derivative is $V^r$. \end{conjecture}

In~\Cite{Hutchings~98}{Hutchings:SingularBraids} Hutchings reduced this conjecture to a certain appealing topological statement and further to a certain combinatorial-algebraic statement about the vanishing of a certain homology group $H^1$ which is probably related to Kontsevich's graph homology complex~\Cite{Kontsevich~94}{Kontsevich:FeynmanDiagrams} (Kontsev ich's $H^0$ is $\calA$, so this is all in the spirit of many deformation theory problems where $H^0$ enumerates infinitesimal deformations and $H^1$ is the obstruction to globalization). Hutchings~\Cite{Hutchings~98}{Hutchings:SingularBraids} was also able to prove t he vanishing of $H^1$ (and hence reprove the Fundamental Theorem) in the simpler case of braids. But no further progress has been made along these lines since~\Cite{Hutchings~98}{Hutchings:SingularBraids}.

\section{Some further directions} \label{sec:FurtherDirections}

We would like to touch on a number of significant further directions in the theory of finite type invariants. We will only say a few words on each of those and refer the reader to the literature for further information.

\subsection{The original ``Vassiliev* perspective} V.A.~Vassiliev came*
to the study of finite type knot invariants by studying the infinite
dimensional space of all immersions of a circle into $\bbR^3$ and the
topology of the ``discriminant*, the locus of all singular immersions*
within the latter space~\Cite{Vassiliev~90,~92}{Vassiliev:CohKnot,
Vassiliev:Book}. Vassiliev studied the topology of the complement of
the discriminant (the space of embeddings) using a certain spectral
sequence and found that certain terms in it correspond to finite type
invariants. This later got related to the Goodwillie calculus and back
to the configuration spaces of Section~\ref{sec:CS}.
See~\Cite{Volic~04}{Volic:TaylorTowers}.

\subsection{Interdependent modifications} The standard definition of
finite type invariants is based on modifying a knot by replacing over
(or under) crossings with under (or over) crossings.
In~\Cite{Goussarov~98}{Goussarov:Modifications} Goussarov generalized
this by allowing arbitrary modifications done to a knot --- just take
any segment of the knot and move it anywhere else in space. The
resulting new ``finite type* theory turns out to be equivalent to the*
old one though with a factor of $2$ applied to the grading (so an
``old* type $m$ invariant is a ``new* type $2m$ invariant and vice
versa). See also~\Cite{Bar-Natan~01, Conant~03}{Bar-Natan:Bracelets,
Conant:OnGoussarov}.

\subsection{$n$-equivalence, commutators and claspers} While little is
known about the overall power of finite type invariants, much is known
about the power of type $n$ invariants for any given $n$.
Goussarov~\Cite{Goussarov~93}{Goussarov:nEquivalence} defined the
notion of $n$-equivalence: two knots are said to be ``$n$-equivalent
if all their type $n$ invariants are the same. This equivalence
relation is well understood both in terms of commutator subgroups of
the pure braid group~\Cite{Stanford~98,
Ng-Stanford~99}{Stanford:ModuloPureBraids, NgStanford:GusarovGroup} and
in terms of Habiro's calculus of surgery over
``claspers*~\Cite{Habiro~00}{Habiro:Claspers} (the latter calculus*
also gives a topological explanation for the appearance of Jacobi
diagrams as in Section~\ref{subsec:Jacobi}). In particular, already
Goussarov~\Cite{Goussarov~93}{Goussarov:nEquivalence} shows that the
set of equivalence classes of knots modulo $n$-equivalence is a
finitely generated Abelian group $G_n$ under the operation of connected
sum, and the rank of that group is equal to the dimension of the space
of type $n$ invariants.

Ng~\Cite{Ng~98}{Ng:Ribbon} has shown that ribbon knots generate an index $2$ subgroup of $G_n$.

\subsection{Polynomiality and Gauss sums} Goussarov~\Cite{Goussarov~98}{Goussarov:PresentedByGauss} (see also~\Cite{Goussarov-Polyak-Viro~00}{GoussarovPolyakViro:VirtualKnots}) found an intriguing way to compute finite type invariants from a Gauss diagram presentation of a knot, showing in particular that finite type invariants grow as polynomials in the number of crossings $n$ and can be computed in polynomial time in $n$ (though actual computer programs are still missing!).

Gauss diagrams are obtained from knot diagrams in much of the same way as Chord diagrams are obtained from singular knots, except all crossings are counted and not just the double points, and certain over/under and sign information is associated with each crossing/chord so that the knot diagram can be recovered from its Gauss diagram. In the example below, we also dashed a subdiagram of the Gauss diagram equivalent to the chord diagram $\eps{width=7mm}{CD123123}$: \[ \eps{width=2.4in}{GaussDiagram} \]

If $G$ is a Gauss diagram and $D$ is a chord diagram we let $\langle D,G\rangle$ be the number of subdiagrams of $G$ equivalent to $D$, counted with appropriate signs (to be precise, we also need to base the diagrams involved and count subdiagrams that respect the basing).

\begin{theorem} \Cite{Goussarov~98, Goussarov-Polyak-Viro~00}{Goussarov:PresentedByGauss, GoussarovPolyakViro:VirtualKnots} If $V$ is a type $m$ invariant then there are finitely many (based) chord diagrams $D_i$ with at most $m$ chords and rational numbers $\alpha_i$ so that $V(K)=\sum_i\alpha_i\langle D_i,G\rangle$ whenever $G$ is a Gauss diagram representing a knot $K$. \end{theorem}

\subsection{Computing the Kontsevich Integral} While the Kontsevich Integral $Z_1$ is a cornerstone of the theory of finite type invariants, it has been computed for surprisingly few knots. Even for the unknot the result is non-trivial:

\begin{theorem} \label{thm:Wheels} (``Wheels*,*
\Cite{Bar-Natan-Garoufalidis-Rozansky-Thurston~00,
Bar-Natan-Le-Thurston~03}{Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling,
Bar-NatanLeThurston:TwoApplications}) The framed Kontsevich integral of
the unknot, $Z_1^F(\bigcirc)$, expressed in terms of diagrams in
$\hat\calB$, is given by $\Omega=\exp_\udot \sum_{n=1}^\infty
b_{2n}\omega_{2n}$, where the `modified Bernoulli numbers' $b_{2n}$ are
defined by the power series expansion $\sum_{n=0}^\infty b_{2n}x^{2n} =
\frac{1}{2}\log\frac{\sinh x/2}{x/2}$, the `$2n$-wheel' $\omega_{2n}$
is the free Jacobi diagram made of a $2n$-gon with $2n$ legs (so e.g.,
$\omega_6=\eps{width=5mm}{6wheel}$) and where $\exp_\udot$ means
`exponential in the disjoint union sense'.
\end{theorem}

Closed form formulas have also been given for the Kontsevich integral of framed unknots, the Hopf link and Hopf chains~\Cite{Bar-Natan-Lawrence~04}{Bar-NatanLawrence:RationalSurgery} and for torus knots~\Cite{Marche~04}{Marche:Computation}.

Theorem~\ref{thm:Wheels} has a companion that utilizes the same element
$\Omega$, the ``wheeling
theorem~\Cite{Bar-Natan-Garoufalidis-Rozansky-Thurston~00,
Bar-Natan-Le-Thurston~03}{Bar-NatanGaroufalidisRozanskyThurston:WheelsWheeling,
Bar-NatanLeThurston:TwoApplications}. The wheeling theorem ``upgrades
the vector space isomorphism $\chi:\calB\to\calA$ to an algebra
isomorphism and is related to the Duflo isomorphism of the theory of
Lie algebras. It is amusing to note that the wheeling theorem (and
hence Duflo's theorem in the metrized case) follows using finite type
techniques from the ``$1+1=2$ on an abacus* identity*
\[ \eps{width=2.6in}{Abacus}. \]

\subsection{Taming the Kontsevich Integral} While explicit calculations are rare, there is a nice structure theorem for the values of the Kontsevich integral, saying that for a knot $K$ and up to any fixed number of loops in the Jacobi diagrams, $\chi^{-1}Z_1(K)$ can be described by finitely many rational functions (with denominators powers of the Alexander polynomial) which dictate the placement of the legs. This structure theorem was conjectured in~\Cite{Rozansky~03}{Rozansky:RationalityConjecture}, proven in~\Cite{Kricker~00}{Kricker:RationalityConjecture} and partially generalized to links in~\Cite{Garoufalidis-Kricker~04}{GaroufalidisKricker:NonCommutativeInvariant}.

\subsection{The Rozansky-Witten theory} One way to construct linear functionals on $\calA$ (and hence finite type invariants) is using Lie algebras and representations as in Section~\ref{subsec:Jacobi}; much of our insight about $\calA$ comes this way. But there is another construction for such functionals (and hence invariants), due to Rozansky and Witten~\Cite{Rozansky-Witten~97}{RozanskyWitten:HyperKahler}, using contractions of curvature tensors on hyper-K\"ahler manifolds. Very little is known about the Rozansky-Witten approach; in particular, it is not known if it is stronger or weaker than the Lie algebraic approach. For an application of the Rozansky-Witten theory back to hyper-K\"ahler geometry check~\cite{HitchinSawon:HyperKahler} and for a unification of the Rozansky-Witten approach with the Lie algebraic approach (albeit at a categorical level) check~\Cite{Roberts-Willeton (in preparation)}{RobertsWillerton:InPreparation}.

\subsection{The Melvin-Morton conjecture and the volume conjecture} The Melvin-Morton conjecture (stated~\Cite{Melvin-Morton~95}{MelvinMorton:Coloured}, proven~\Cite{Bar-Natan-Garoufalidis~96}{Bar-NatanGaroufalidis:MMR}) says that the Alexander polynomial can be read off certain coefficients of the coloured Jones polynomial. The Kashaev-Murakami-Murakami volume conjecture (stated~\cite{Kashaev:HyperbolicVolume, MurakamiMurakami:SimplicialVolume}, unproven) says that a certain asymptotic growth rate of the coloured Jones polynomial is the hyperbolic volume of the knot complement.

Both conjectures are not directly about finite type invariants but both have ramifications to the theory of finite type invariants. The Melvin-Morton conjecture was first proven using finite type invariants and several later proofs and generalizations (see~\cite{Bar-Natan:VasBib}) also involve finite type invariants. The volume conjecture would imply, in particular, that the hyperbolic volume of a knot complement can be read from that knot's finite type invariants, and hence finite type invariants would be at least as strong as the volume invariant.

A particularly noteworthy result and direction for further research is Gukov's~\cite{Gukov:APolynomial} recent unification of these two conjectures under the Chern-Simons umbrella (along with some relations to three dimensional quantum gravity).

\section{Beyond knots} \label{sec:BeyondKnots} For the lack of space we
have restricted ourselves here to a discussion of finite type
invariants of knots. But the basic ``differentiation* idea of*
Section~\ref{sec:intro} calls for generalization, and indeed it has been
generalized extensively. We will only make a few quick comments.

Finite type invariants of homotopy links (links where each component is
allowed to move across itself freely) and of braids are extremely well
behaved. They separate, they all come from Lie algebraic
constructions~\Cite{Bar-Natan~95,~96, Lin~97}{Bar-Natan:Homotopy,
Lin:Expansions, Bar-Natan:Braids} and in the case of braids, step by
step integration as in Section~\ref{subsec:StepByStep}
works~\Cite{Hutchings~98}{Hutchings:SingularBraids} (for homotopy links
the issue was not studied).
Finite type invariants of 3-manifolds and especially of integral and
rational homology spheres have been studied extensively and the picture
is nearly a complete parallel of the picture for knots. There are
several competing definitions of finite type invariants (due to
\Cite{Ohtsuki~96}{Ohtsuki:IntegralHomology} and then
\Cite{Goussarov~99, Garoufalidis~96,
Garoufalidis-Goussarov-Polyak~01}{Goussarov:3Manifolds,
Garoufalidis:3ManifoldsI, GaroufalidisGoussarovPolyak:Clovers} and
more), and they all agree up to regrading. There are weight systems and
they are linear functionals on a space $\calA(\emptyset)$ which is a
close cousin of $\calA$ and $\calB$ and is related to Lie algebras and
hyper-K\"ahler manifolds in a similar way. There is a notion of a
``universal* invariant, and there are several constructions (due to*
\Cite{Le-Murakami-Ohtsuki~98, Le~96}{LeMurakamiOhtsuki:Universal,
Le:UniversalIHS} and then
\Cite{Bar-Natan-Garoufalidis-Rozansky-Thurston~02, 02, 04,
Kuperberg-Thurston~99}{Bar-NatanGaroufalidisRozanskyThurston:Aarhus,
KuperbergThurston:CutAndPaste}), they all agree or are conjectured to
agree, and they are related to the Chern-Simons-Witten theory.

Finite type invariants were studied for several other types of topological objects, including knots within other manifolds, higher dimensional knots, virtual knots, plane curves and doodles and more. See~\cite{Bar-Natan:VasBib}.

\begin{thebibliography}{BNT}

\item[] The reference~\cite{Bar-Natan:VasBib} is an extensive bibliography of finite type invariants, listing more than \ifemp{550}{500 further} references. \ifemp{To save space, references for this article that appear in~\cite{Bar-Natan:VasBib} are only cited above stating the authors' full last name(s) and approximate year of publication.}{}

\vskip 3mm

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{\em Vassiliev knot invariants and Chern-Simons perturbation theory to all orders,} Comm.{} Math.{} Phys.{} {\bf 187} (1997) 261--287, \arXiv{q-alg/9603010}.

}

\ifemp{}{% 1991 \bibitem[BN1]{Bar-Natan:Thesis} D.~Bar-Natan,

{\em Perturbative aspects of the Chern-Simons topological quantum field theory}, Ph.D.{} thesis, Princeton Univ., June 1991.

}

\ifemp{}{% 1995 \bibitem[BN2]{Bar-Natan:OnVassiliev} D.~Bar-Natan,

{\em On the Vassiliev knot invariants,} Topology {\bf 34} (1995) 423--472.

}

\ifemp{}{% 1995 \bibitem[BN3]{Bar-Natan:Homotopy} D.~Bar-Natan,

{\em Vassiliev homotopy string link invariants,} Jour.{} of Knot Theory and its Ramifications {\bf 4} (1995) 13--32.

}

\ifemp{}{% 1997 \bibitem[BN4]{Bar-Natan:NAT} D.~Bar-Natan,

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}

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drorbn/\linebreak[0]LOP.html\#\linebreak[0]Computations}. }

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}

\ifemp

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

\ifemp

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}