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	\begin{document}

	\begin{center}
	{\bf Link Invariants of Finite Type and \\}
	{\bf  Perturbation Theory \\}
	{\small  {\it Lett.\ Math.\ Phys.\ }{\bf 26} (1992) 43 - 51\\}
	\vspace{0.5cm}
	{\em John C. Baez\\}
	\vspace{0.3cm}
	{\small Department of Mathematics \\
	Wellesley College\\
	Wellesley, Massachusetts 02181\\
	(on leave from the University of California at Riverside)\\ }
	\vspace{0.3cm}
	{\small  July 13, 1992}
	\vspace{0.5cm}	
	\end{center}

\begin{abstract}
The Vassiliev-Gusarov link invariants of finite type are known to be
closely related to perturbation theory for Chern-Simons theory.  In
order to clarify the perturbative nature of such link invariants, we
introduce an algebra $V_\infty$ containing
 elements $g_i$ satisfying the usual braid group relations and
elements $a_i$ satisfying $g_i - g_i^{-1} = \eps a_i$, where $\eps$ is
a formal variable that may be regarded as measuring the failure of
$g_i^2$ to equal 1.   Topologically, the elements $a_i$ signify
intersections.  
We show that a large class of link invariants of finite type are in
one-to-one correspondence with homogeneous Markov traces on 
$V_\infty$.     
We sketch a possible application of link invariants of finite type to
a manifestly diffeomorphism-invariant perturbation theory for quantum
gravity in the loop representation.  
\end{abstract}

\section{Introduction}


The manner in which the braid group $B_n$ takes the place of the
symmetric group $S_n$ in the representation theory of quantum groups
is by now well known.
Recall that the braid group $B_n$ has a presentation with generators
$s_i$, $1 \le i < n$, and relations
\ba    s_i s_j &=& s_j s_i \qquad\qquad\qquad |i-j| > 1, \nonumber\cr
s_is_{i+1}s_i &=& s_{i+1}s_i s_{i+1} .\nonumber\ea
The symmetric group $S_n$ is the quotient of $B_n$ by the further
relations $s_i^2 = 1$.  
If $G$ is a semisimple Lie group, then the corresponding quantized
enveloping algebra
$U_q$\g\hskip 0.5 em   is a deformation of the universal enveloping
algebra $U$\g\hskip 0.5 em  as Hopf algebras.     Given any
representation of the group $G$,  
there is a natural action of the group algebra 
$\CS_n$ as intertwining operators on the representation $E^{\tensor n}$.  
Similarly, given any representation $E$ of $U_q$\g, there
is a natural action of $\CB_n$
as intertwining operators on $E^{\tensor n}$. 

Naively, one might be led to hope that $\CB_n$ is 
a kind of deformation of $\CS_n$.  While this is not true according to
the standard definition - after all, 
$\CB_n$ is infinite-dimensional while $\CS_n$ is finite-dimensional -
we show here that some sense can be made of this idea.  
Roughly speaking, one can form an algebra $V_n$ over $\C[\eps]$, where
$\eps$ is a formal variable, by adjoining to $\CB_n$ elements 
$a_i$, $1 \le i < n$, such that
\[                  s_i - s_i^{-1} = \eps a_i  .\]
The parameter $\eps$ should be thought of as measuring the
failure for $s_i^2$ to equal $1$.  Setting 
$\eps$ equal to any nonzero constant gives the algebra $\CB_n$, while
setting $\eps = 0$ gives an algebra containing $\CS_n$.   
More generally, for any $d \ge 0$,
the quotient algebra $V_n/\langle \eps^{d+1} \rangle$
should be regarded as a $d$th-order perturbative approximation to $\CB_n$.

These quotients are closely related to the theory of link invariants of
finite type, as developed by Vassiliev, Gusarov, Birman, Lin,
Bar-Natan, and others \cite{Bar-Natan,BL,Gus,Lin,Vass}.  
The basic idea here is that one can canonically extend an invariant
$\L$ of
oriented links to an invariant of generalized links
admitting nice self-intersections (transverse double points)
by means of the rule
\[      \L(L_+) - \L(L_-) = \eps \L(L_\times) ,\]             
Here $L_+$ is as in Figure 1, $L_-$ is as in Figure 2, and $L_\times$
is as in Figure 3, with the strands oriented so as to be pointing
{\it downwards}.  
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Figure 1.
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Figure 2.
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Figure 3.   
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Those invariants vanishing on
all generalized links with more than $d$ self-intersections are said
to be of degree $d$.   The space of link invariants
of degree $d$ can be regarded as a
$d$th-order approximation to the dual of the space with basis given by
isotopy classes of links.
If one takes one of the $\C(q)$-valued link invariants 
derived from quantum group representations by the procedure of
Turaev \cite{Tur}, sets $q = \exp(\eps)$ to obtain a formal power
series in $\eps$, and takes the coefficient
of $\eps^d$, one obtains a link invariant of degree $d$.  
More generally, we show that there is a one-to-one correspondence
between a large class of link invariants of degree $d$
and Markov traces $\tau \maps V_\infty \to \C[\eps]$ that are
``homogeneous of degree $d$'' in a certain sense.  Such Markov traces 
may be thought of as defined on $V_\infty/\langle \eps^{d+1}\rangle$.
 
The connection between link invariants of finite type 
and {\it physical} perturbation theory is presently clearest
in the context of Chern-Simons theory with 
semisimple gauge group $G$.  Here the action is given by
\[   S = {k\over 4\pi} \int_{S^3} \tr (A \wedge dA + {2\over 3}A \wedge A
\wedge A) ,\]
where $A$ is a $G$-connection and the level $k \ge 0$ is an integer.   
From the work of Witten \cite{Witten}
and others it is clear that, for example, the Jones polynomial $V_L(q)$
may be obtained from the vacuum
expectation values of Wilson loops in Chern-Simons theory with $G =
SU(2)$ and $q = \exp(2\pi i/(k+2))$.
The coefficient of the $\eps^d$ term of the
Jones polynomial, an invariant of finite degree, should thus be
calculable by a $d$th-order perturbation expansion in Chern-Simons
theory.   This has been pursued by Cotta-Ramusino {\it et al}, Smolin,
and others \cite{Guad,Smolin1}.
In particular, the relation to knot invariants of finite type
has been studied by Bar-Natan \cite{Bar-Natan}, who  
dealt with an arbitrary classical Lie group,
and by abstraction obtained
a general combinatorial scheme for constructing knot invariants of
finite type. 

Link invariants of finite
type may also be expected to play a role in a novel
perturbation theory for $4$-dimensional quantum
gravity.   In the loop representation of quantum gravity developed by
Rovelli  and Smolin \cite{RS}, states are described by linear
combinations of isotopy classes of framed unoriented links (or tangles), 
possibly admitting self-intersections. 
We briefly comment on the relation
between Chern-Simons perturbation theory, link invariants of finite
type, and perturbative quantum gravity in the final section of this paper.

The author thanks Dror Bar-Natan, J\'ozef Przytycki, and
Stephen Sawin for useful discussions concerning link invariants of
finite type, Allen Knutson for catching many errors,
and Micheal Weiss for help with preparing the figures.

\eject
\section{The Vassiliev Algebra}

Let the {\it generalized braid monoid}, $GB_n$, denote 
the monoid with generators 
$g_i, g_i^{-1}, a_i$, $1 \le i < n$, and relations
\ba    [g_i, g_j] &=& \;[a_i, a_j]\; = \;[a_i, g_j]\; = 0  
\qquad\qquad\qquad |i-j| > 1, \nonumber\cr
g_ig_i^{-1} &=& g_i^{-1}g_i \;=\; 1  , \nonumber\cr
a_i g_i &=& g_i a_i  ,\nonumber\cr
g_ig_{i+1}g_i &=& g_{i+1}g_i g_{i+1}, \nonumber\cr
g_{i+1}a_ig_{i+1}^{-1} &=&  g_i^{-1}a_{i+1}g_i  ,\nonumber\cr
g_{i+1}^{-1}a_ig_{i+1} &=&  g_i a_{i+1}g_i^{-1} .\nonumber\ea
In pictures of generalized braids, $g_i$ represents a right-handed
crossing and $g_i^{-1}$ represents a left-handed crossing of the $i$th and
$(i+1)$st strands, as in Figures 1 and 2.  The element $a_i$ represents an
intersection of the $i$th and $(i+1)$ strands, as in Figure 3. The
relations above express topological facts about generalized braids,
which admit intersections as well as crossings; the reader is strongly
encouraged to draw these relations.
The generalized
braid monoid appears in the work of Br\"ugmann, Gambini and Pullin [7,9]
on quantum gravity; see also [11].

Let $\CGB_n$ denote the monoid
algebra of the generalized braid monoid.  
We define the {\it Vassiliev algebra}, $V_n$, to be the
quotient of $\CGB_n \tensor \C[\eps]$ by the ideal generated by the
elements 
\[g_i - g_i^{-1} =  \eps a_i  .\]
It is clear that there is a homomorphism 
\[     v \maps \CB_n \to V_n \] 
given by $s_i \mapsto g_i$. 
The basic properties of this homomorphism are as
follows. 

\begin{lemma}\label{lem1}\et  Let $\C(\eps)$ denote the algebra of Laurent
polynomials in $\eps$.  Then 
\[v \tensor 1 \maps \CB_n \tensor
\C(\eps) \to V_n \tensor_{\C[\eps]} \C(\eps)\]
is an isomorphism.
\end{lemma}

Proof - The inverse is 
given by $g_i \mapsto s_i$, $a_i \mapsto \eps^{-1}(s_i - s_i^{-1})$.
\qed

\begin{corollary} \label{cor1}\et   Let $j \maps V_n \to V_n/ \langle
\eps - x\rangle$ denote the quotient map.  The map
\[        j\circ v \maps \CB_n \to V_n /\langle \eps - x\rangle \]
is an isomorphism if $x \in C$ is nonzero, while if $x = 0$ it factors
through $\CS_n$.
\end{corollary}

Proof - The composite $j\circ v$ is an isomorphism for $x \ne 0$ by
Lemma \ref{lem1}, while if $x = 0$, $s_i^2 = 1$ in $V_n/\langle
\eps\rangle$, so $j\circ v$ factors through $\CS_n$.  \qed

\begin{corollary} \label{cor2}\et  The homomorphism $v \maps \CB_n \to
V_n$ is one-to-one.  \end{corollary}

Proof - This is immediate from Lemma \ref{lem1}.  \qed

We conclude this section with a word on the universal role the
Vassiliev algebra plays in the context of braided tensor categories.
It is clear that, given any object $E$ in a $\C[\eps]$-linear strict
braided monoidal category, if $R \equiv R^{-1} \bmod
\eps$, where $R \maps E \tensor E \to E \tensor E$ is the
braiding, then there is a canonical action of $V_n$ as endomorphisms of
$E^{\tensor n}$.  In particular, this applies to the category of
quantum group representations, where we write $q = \exp(\eps)$.

\section{Link Invariants of Finite Type}

By a link invariant we will always mean an ambient isotopy invariant
of oriented links.
It is easy to see that any $\C$-valued link invariant $\L$ uniquely
extends to a $\C(\eps)$-valued invariant of generalized
links admitting transverse double points, which we also call
$\L$, by means of the rule 
\[       \L(L_+) - \L(L_-) = \eps\L(L_\times) ,\]
where $L_+$, $L_-$, and
$L_\times$ denote link diagrams with a right-handed crossing, a
left-handed crossing, and an intersection, respectively, at a given
point, the rest of the diagrams being the same.   We
define a $\C$-valued
link invariant to be of {\it degree $d$} if it vanishes on all generalized
links with $d+1$ or more self-intersections.  
A link invariant of degree $d$ for some $d$ is said to be of {\it
finite type}.   

For all $n$ there are algebra inclusions $V_n \hookrightarrow V_{n+1}$
and $\CB_n \hookrightarrow \CB_{n+1}$.   
Let $V_\infty$ and $\CB_\infty$ denote the inductive limits of the
algebras $V_n$ and $\CB_n$, respectively, and let
\[    v \maps \CB_\infty \to V_\infty \] 
denote the inductive limit of the maps $v \maps \CB_n \to V_n$.
We define a {\it Markov
trace} on $V_\infty$ to be a $\C[\eps]$-linear map $\tau \maps V_\infty
\to E$, where $E$ is some $\C[\eps]$-module, satisfying
\[           \tau(xy) = \tau(yx)  \]
for all $x,y \in V_\infty$, and for some fixed $z \in \C$
\[           \tau(g_n^{\pm 1} x) = z\tau(x)  \]
for all $x \in V_n \subset V_\infty$.   
A similar definition is standard for Markov traces $\tr \maps
\CB_\infty \to \C$.  We say that a Markov trace $\tau \maps 
V_\infty \to \C[\eps]$ is {\it homogeneous of degree $d$} if for every
$x \in \CB_\infty$, $\tau(v(x))$ is homogeneous of degree $d$ as a
polynomial in $\eps$.  

For any braid $x \in B_n$, let $\hat x$ denote its closure.  Of
course, given $x \in B_\infty$, $\hat x$ depends on a choice of 
$n$ such that $x \in B_n$.   Given a link $L$, let $L \cup \circ$
denote the distant union of $L$ with the unknot.  

\begin{theorem}\et There is a one-to-one correspondence between
$\C$-valued link invariants $\L$ of degree $d$ 
such that for some $z \ne 0$ and all links $L$,  
\be        \L(L \cup \circ) = z^{-1} \L(L),       \label{circ}  \ee
and Markov traces $\tau \maps V_\infty \to \C[\eps]$ 
that are homogeneous of degree $d$.
The invariant $\L$ determines the trace $\tau$, and conversely, by the
property that 
\be     \tau(v(x)) = \eps^d z^{n-1}\L(\hat x)      \label{CORR} \ee
for $x \in B_n$.  
\end{theorem} 

Proof - 
Let $E$ be a vector space, and suppose $\L$ is a $E$-valued link
invariant satisfying equation (\ref{circ}).   Then by 
Markov's theorem \cite{Birman} there is a 
Markov trace $\tr \maps \CB_\infty \to E$, given by
\be               \tr(x) = z^{n-1} \L(\hat x)    \label{corr}  \ee
for all $x \in B_n$.  Note in particular that $\tr$ is well-defined
because if $y \in \CB_{n+1}$ is the image of $x \in \CB_n$ under the
inclusion $\CB_n \hookrightarrow \CB_{n+1}$, then
\[  \tr(y) = z^n\L(\hat y) = z^n\L(\hat x \cup \circ) =
z^{n-1}\L(\hat x) = \tr(x) .\]
Moreover $\tr$ is a trace because $\widehat{xy} = \widehat{yx}$ for
all $x,y \in B_n$, and $\tr$ has the Markov property:
\[   \tr(s_n^{\pm 1}x) = z^n \L((s_n^{\pm 1}x)\widehat{\;})
= z^n \L(\hat x) = z\, \tr(x)  .\]
In fact, Markov's theorem implies that equation (\ref{corr})
gives a one-to-one correspondence
between $E$-valued link invariants satisfying equation (\ref{circ})
and Markov traces $\tr \maps \CB_\infty \to E$.

Now let $\L$ be a $\C$-valued link invariant of degree $d$ satisfying
equation (\ref{circ}).   Then there is a unique Markov trace
$\tr \maps \CB_\infty \to \C$ given by equation (\ref{corr}).  
We claim that there exists a Markov trace 
$\tau_0 \maps V_\infty \to \C(\eps)$ such that
\[        \eps^d \tr = \tau_0   v .\]
To see this, note that by Lemma \ref{lem1} there is a Markov trace
$\tilde\tau_0 \maps V_\infty \tensor_{\C[\eps]} \C(\eps)
\to \C(\eps)$ such that
\[          \eps^d (\tr \tensor 1) = \tilde\tau_0 (v \tensor 1)\]
as maps from $\CB_\infty \tensor \C(\eps)$ to $\C(\eps)$. 
Let $\tau_0$ denote the restriction of $\tilde \tau_0$ to 
$V_\infty$, which may be regarded as a subalgebra of $V_\infty
\tensor_{\C[\eps]} \C(\eps)$.  It follows that $\eps^d \tr = 
\tau_0 v$, as desired.

Note that $\tau_0(x) = c\eps^{d-\ell}$ for some $c \in \C$ if $x \in
V_\infty$ is a product of $\ell$ elements of the form $a_i$ and
arbitrarily many of the form $g_i$.   Moreover, if $\ell > d$
then $\tau_0(x)$ vanishes, since $\L$ is of degree $d$.  It follows that 
\[        \tau_0(x) = \sum_{i=0}^d c_i \eps^i,  \]
with $c_i \in\C$, if $x$ is a product of elements of the form $a_i$
and $g_i$.   It follows that $\tau_0$ factors through a map 
$\tau \maps V_\infty \to C[\eps]$.
It is easy to check that $\tau_0$, hence $\tau$, is a Markov trace
that is homogeneous of degree $k$.  Moreover,
equation (\ref{CORR}) holds by construction.

Conversely, suppose that $\tau \maps V_\infty \to C[\eps]$ 
is a Markov trace homogeneous of degree $d$.  We may define a Markov trace 
$\tr \maps \CB_\infty \to \C$ by 
\[        \tau v = \eps^d \tr .\]
Associated to this trace there is a link invariant $\L$ satisfying equation
$\ref{circ}$, given by equation (\ref{corr}).  We claim that $\L$ is
an invariant of degree $d$. 
For this, we need an analog of Alexander's theorem for
generalized links.   Given an element $x \in GB_n$, we may form a
generalized link $\hat x$, the {\it closure} of $x$, in a manner
analogous to the usual closure of a braid.  

\begin{lemma}\et  For every generalized link $L$, for some $n$
there is an element $x \in GB_n$ such that $\hat x$ is ambient
isotopic to $L$.  \end{lemma}

Proof - We omit the proof, as it is similar to the usual
proof of Alexander's theorem \cite{Birman}, but would be quite
long with all the details included.   \qed

Now let $L$ be a generalized link with $\ell$ self-intersections, and
let $x_0 \in GB_n$ be such that $\hat x_0$ is ambient isotopic to $L$.  
Let $x_1$ be the image of $x_0$ in $V_\infty$.
Then 
\[     x_1 = y_1a_{i_1}y_2a_{i_2} \cdots a_{i_\ell}y_{\ell + 1}  \]
where the elements $y_i$ are products of elements $g_j, g_j^{-1}$.
Define $x \in \CB_\infty$ by
\[     x = y_1(g_{i_1} - g_{i_1}^{-1})y_2
 \cdots (g_{i_\ell} - g_{i_\ell}^{-1})y_{\ell + 1} . \]
Note that
\[         \eps^d \tr(x) = \tau(v(x))  = \eps^\ell 
\tau(x_1) . \]
Since $\tr(x) \in \C$ and $\tau(x_1) \in \C[\eps]$, so
if $\ell > d$ we must have $\tr(x) = 0$.
By construction,
\[      \L(L) = z^{1-n}\tr(x) ,\]
so it follows that $\L(L) = 0$.

Now let us show that the invariant $\L$ uniquely determines the trace
$\tau$, and vice versa, by equation (\ref{CORR}).   Since every link
is the closure of some braid, $\L$ is determined by $\tau$. 
Conversely, $\L$ determines $\tau$ on the image of $v$, since $v$ is
one-to-one by Corollary \ref{cor2}.   It then follows by the
$\C[\eps]$-linearity of $\tau$ that $\tau$ is determined on all of
$V_\infty$, since $\eps a_i = s_i - s_i^{-1}$.  
 \qed

The reader may be puzzled by the fact that every link invariant of
degree $d$ is obviously of degree $d+1$, while a normalized Markov
trace $\tau \maps V_\infty \to \C[\eps]$ that is homogeneous of degree
$d$ is definitely {\it not} homogeneous of degree $d+1$.  
The point is that $\eps \tau$ will be a normalized Markov
trace homogeneous of degree $d+1$.  

Note that to reconstruct the link invariant coming from a normalized
Markov trace  $\tau \maps V_\infty \to \C[\eps]$ that is homogeneous
of degree $d$, it suffices to know the composite of $\tau$ with the
quotient map $\C[\eps] \mapsto \C[\eps]/\langle \eps^{d+1} \rangle$,
which may be regarded as a Markov trace
\[      \tau \maps V_\infty/\langle \eps^{d+1}\rangle \to  
\C[\eps]/\langle \eps^{d+1} \rangle .\]

As an illustration of the theorem, 
let $\tr_0 \maps \CB_\infty \to 
\C(q)$ be one of the traces obtained from quantum group
representations by Turaev's procedure.  Writing $q = \exp(\eps)$, we
regard $\tr_0$ as having values in $\C\fps$.  
For some invertible $z \in \C\fps$, 
\[     \tr_0(s_n^{\pm 1} x) = z\,\tr_0(x)  \]
for all $x \in \CB_n$.  Since $z \notin \C$, the theorem above is not
directly applicable.  However, we may define a new Markov trace $\tr \maps
\CB_\infty \to \C\fps$ with 
\[      \tr(s_n^{\pm 1} x) = \tr(x)  \]
for all $x \in \CB_n$ by setting
\[      \tr(x) = z^{m-n} \tr_0(x)  ,\]
for $x \in B_n$, where $m$ denotes the number of components of $\hat x$.
There is a unique Markov trace $\tau \maps V_\infty \to \C\fps$ such that
\[    \tau v = \tr .\]
Moreover, we may write
\[       \tau = \sum_{d= 0}^\infty \tau_d  \]
where $\tau_d \maps V_\infty \to \C[\eps]$ is a Markov trace
homogeneous of degree $d$, with $\tau_d(g_n^{\pm 1}x) = \tau_d(x)$
for all $x \in V_n$.  By the theorem, each such trace gives a
link invariant $\L_d$ of degree $d$, such that
\[            \L_d(\hat x) = \eps^{-d} \tau_d(v(x)) \]
for $x \in B_n$.   

It is also important that the space of Markov traces $\tau \maps
V_\infty \to \C[\eps]$ that are homogeneous of degree $d$ is
finite-dimensional.  This may be seen by graph-theoretic reasoning
along the lines of Birman and Lin \cite{BL}, Bar-Natan
\cite{Bar-Natan}, and Stanford \cite{Stanford}. 
Suppose, for example, that we fix the value of $z$.
Then if $d = 0$, 
$\tau$ is determined by its value on $1$.   If $d = 1$,
$\tau$ is determined by its value on $1, a_1,$ and $a_1s_1$.  If
$d = 2$, $\tau$ is determined by its value on $1, a_1, a_1s_1, 
a_1^2, a_1^2s_1, a_1a_2, a_1a_2s_1s_2, a_1a_3, a_1a_3s_1,$
and $a_1a_3s_1s_3$.  

\section{Connections to Physics}

The Vassiliev algebra formalism makes clear that link invariants of
finite type arise from a sort of ``topological perturbation
theory.''  We have pointedly used the symbol $\eps$ in our paper,
instead of the suggestive letter $h$ (common
in the quantum group literature), because the physical interpretation
of this sort of perturbation theory is an
interesting issue.   

In $SU(n)$ Chern-Simons theory one makes the identification
\[     \eps = {2\pi i\over k + n  } . \]
The limit $\eps \to 0$ thus corresponds to $k
\to \infty$, which may regarded either as a weak coupling limit or
as a classical limit.    In the weak coupling approach \cite{Witten} we
may write $A = A_0 + k^{-1/2} B$, where
$A_0$ is a flat connection, and obtain
\[   S = kI + {1\over 8\pi} \int \eps^{abc}
 \tr (B_a D_b B_c + {2\over 3}k^{-\hf} B_a[B_b,B_c]),\]
where $I$ is the Chern-Simons invariant of $A_0$ and $D$ denotes the
covariant derivative with respect to $A_0$.    Here we see that the
$k \to \infty$ limit is closely related to a deformation of the Lie
algebra {\su}$(n)$ to an abelian Lie algebra by scaling the bracket.  
Alternatively, since $k$ appears
where one would expect a factor of $\hbar^{-1}$ in the partition function
\[   Z = \int {\cal DA} \exp\left( {ik\over 4\pi} \int \tr (A \wedge
dA + {2\over 3}A \wedge A \wedge A\right), \]
one may also regard $k \to \infty$ as a classical limit.  This is
consistent with the weak coupling interpretation, of course, since the
classical solutions of Chern-Simons theory are flat connections.  

In quantum gravity the two obvious limits to consider are the
classical limit $\hbar \to 0$ and the $G \to 0$ limit, where $G$ is
Newton's gravitational constant.  The latter has been considered in the
loop representation of Euclidean quantum gravity by Smolin
\cite{Smolin2}, with the aim of developing a new perturbation theory
for quantum gravity which is manifestly diffeomorphism-invariant at
every order, as opposed to perturbation about a flat background spacetime.
In this approach an $SU(2)$ connection is
a key dynamical variable \cite{Ash}, and Smolin shows that the role of
$G$ is to scale 
the Lie bracket in {\su}$(2)$.   The analogy with the $k \to \infty$
limit of Chern-Simons theory is no accident, since  
states of quantum gravity may be obtained
from Wilson loops in $SU(2)$ Chern-Simons theory \cite{Kodama}.   
In this connection, the
extension of the Jones polynomial to generalized links by Br\"ugmann,
Gambini and Pullin \cite{BGP} is quite intriguing.

It is thus natural to
hope that link invariants of finite type will play an important role
in the $G \to 0$ limit of quantum gravity, or similar perturbation
expansions in more general tangle field theories \cite{Baez}.  
Roughly, we may expect that the true physical Hilbert space $\H$
is an inverse limit of spaces $\H_d$, where two states in $\H$ are
identified in $\H_d$ if they cannot be distinguished by link (or tangle) 
invariants of degree $d$.    

{\it Note added in proof:} Joan Birman has simultaneously and independently
discovered the results of this paper.  Birman also proves that the
relations in $GB_n$ are as stated.  Her work will appear in ``New points
of view in knot theory,'' {\it Bull.\ A.M.S.}.


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