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	\begin{center}
	{\bf Quantum Gravity and the Algebra of Tangles\\}
         {\small  Jour.\ Class.\ Quantum Grav.\ {\bf 10} (1993) 673 - 694\\}
	\vspace{0.5cm}
	{\em John C. Baez\\}
	\vspace{0.3cm}
	{\small Department of Mathematics \\
	University of California\\ 
        Riverside CA 92521\\}
	\vspace{0.3cm}
	{\small May 4, 1992}
	\vspace{0.5cm}	
	\end{center}

\begin{abstract}  In the loop representation of
quantum gravity in 3+1 dimensions, there is a space of
physical states consisting of invariants of links in $S^3$.  
The correct inner product on
this space of states is not known, or in other words, the
$\ast$-algebra structure of the algebra of observables has not been
determined.  In order to address this issue, we work instead
with quantum gravity on $D^3$, and work with a space
$\H$ spanned by tangles.
A certain algebra $\T$, the ``tangle algebra,'' acts as operators on
$\H$.  The ``empty link''
$\psi_0$, corresponding to the class of the empty set, is
shown to be a cyclic vector for $\T$. 
We construct inner products on quotients of $\H$ from link invariants,
show that these quotients are representations of $\T$, and
calculate the $\ast$-algebra structures of $\T$ in these
representations.   In particular, taking the link invariant to be
the Jones polynomial (or more precisely, Kauffman bracket), we obtain
the inner product for states of quantum gravity arising from $SU(2)$
Chern-Simons theory.  
\end{abstract}

\section{Introduction}

In the loop representation of nonperturbative quantum
gravity in 3+1 dimensions, due to
Ashtekar, Rovelli and Smolin \cite{Ashtekar,RS,Smolin}, one
begins with a ``kinematical'' state space described in terms
of loops, and then seeks ``physical'' states, that is, those
that satisfy the Hamiltonian and spatial diffeomorphism
constraints.  A large space of physical states is known, namely,
the space of link invariants,
which is dual to the space spanned by isotopy classes of
links.  While there is much to be done towards finding the
full space of physical states, and clarifying
technical issues related to regularization and operator ordering,
the main ``practical'' problem is that the inner product on the 
space of link invariants 
is not known; it is presently merely a vector space, not a Hilbert space. 
Indeed, it is not even known which link invariants
have finite norm relative to the physical inner product.

This is important because an inner product is essential in the
probabilistic interpretation of a quantum theory.  Alternatively, one
could say that the problem is the lack of an adjoint on the algebra of 
linear operators on the physical state space.  Recall that an
algebra $A$ is a {\it $\ast$-algebra} if there is a map $\ast \maps A
\to A$ such that
\[    (a^\ast)^\ast = a,\quad (\lambda a)^\ast = \overline \lambda a^\ast,
\quad (a + b)^\ast = a^\ast + b^\ast, \quad (ab)^\ast = b^\ast a^\ast .\]
 Until an operator algebra 
has been given the structure of a $\ast$-algebra, one can identify
neither the observables (self-adjoint elements) nor the physical states 
(functionals $\psi \maps A \to \C$ such that $\psi(1) = 1$ and
$\psi(a^\ast a) \ge 0$ for all $a \in A$).   Thus the algebra of
operators for a physical system should be a $\ast$-algebra.
For reasons of mathematical convenience and physical principle, 
it is usually assumed that the algebra of observables is a C*-algebra.  

Typically the inner product in a quantum field theory is constrained
by the fact that physical symmetry groups, in particular the Poincar\'e
group, should act unitarily.    This is of no help in quantizing gravity,
except perhaps in the asymptotically flat case, where at least
classically the Poincar\'e group acts.  One may instead seek physical
observables for quantum gravity and constrain the inner product by
requiring that they be self-adjoint, but presently,
practically no operators on the physical state space are known to 
correspond to observables.   The reason is that since the physical 
states satisfy the Hamiltonian and diffeomorphism
constraints, such observables would be diffeomorphism-invariant, 
i.e.,  independent of any choice of spacetime
coordinates.  In terms of the loop
representation, apart from the problem of identifying the physical 
meaning of operators on the space of link invariants, 
there is the challenge of finding a tractable computational framework
for such operators.

While it is difficult to describe enough operations on isotopy classes
of links to obtain a powerful algebra of operators on the space of
physical states for quantum gravity on $S^3$, the situation is much
improved if instead we split $S^3$ into two copies of the closed ball
$D^3$ and work with isotopy classes of
 tangles rather than directly with links.  
 Roughly speaking, a tangle is a 1-dimensional submanifold of
$D^3$, consisting of links in the interior and arcs connecting points
on the boundary; two tangles are isotopic if one can be
transformed into the other by a continuous family of diffeomorphisms of
$D^3$ that equal the identity on the boundary, $S^2$.   
The goal of this paper is twofold.  First, 
we describe an algebra of operators
on the space spanned by isotopy classes of tangles, the 
``tangle algebra,'' which provides a natural framework for computations.
Then we show how states of quantum gravity on $S^3$ corresponding to link
invariants determine $\ast$-algebra structures on the tangle algebra and
representations of this algebra on Hilbert spaces.
In this we rely heavily on the work of Jones \cite{Jones} and others,
who have discovered 
profound relationships between link invariants and the theory of
C*-algebras.  It is natural to hope that these relationships will
contribute to an understanding of the inner product problem in quantum
gravity.  We wish to emphasize, however, that the present paper is only 
a first step in this direction.

While somewhat technical, it is important here to
mention the issue of equipping tangles with framings and orientations.
Rovelli and Smolin's original construction worked with unoriented links.
 More recent work of Br\"ugmann, Gambini and Pullin \cite{BGP}
suggests that framed links are required to deal with
regularization issues.
Their work is closely related to the need for framings
in Chern-Simons theory \cite{Witten}.  Certainly, to
make contact with the Reshetikhin-Turaev theory of link invariants
\cite{Turaev,RT}, framings should be taken into account.
In addition to framings, orientations may be required in theories of
gravity coupled to matter.    Here, however, in view of applications to
pure gravity, we work with framed unoriented tangles.

In Section 2 we begin by studying a space $\widetilde \H$
having as a basis isotopy classes of framed unoriented tangles.
We develop a computational framework for handling
symmetries and other natural operators on the space $\widetilde \H$.
The group of orientation-preserving diffeomorphisms of $S^2$ acts
as symmetries on $\widetilde \H$.  One may take the quotient by
almost all of this group to obtain a ``reduced'' space $\H$.
The remaining symmetries are described by a discrete group, the
framed braid group, which acts on tangles by braiding the strands
that meet the boundary of $D^3$.   Information obtained at the level
of the reduced space can easily be transferred back to the original
space $\widetilde \H$.   In particular, any inner product on $\H$ that is
preserved under the framed braid group action gives rise to a unitary
representation of $\Diff^+(S^2)$ on $\widetilde \H$.

The action of the
framed braid group, together with certain
``creation'' and ``annihilation'' operators that create and close off
strands meeting the boundary, generates an algebra acting on $\H$,
the tangle algebra.
In a quite precise sense, the tangle algebra
consists of all operations on tangles that can be done by manipulating
them only along the boundary of $D^3$.  The whole space $\H$ may be
built up from the vector $\psi_0$ corresponding to the empty link
(the empty set regarded vacuously as a link) by applying operators in the
tangle algebra.    Thus the empty link plays a role somewhat like that
of the vacuum in ordinary quantum field theory, although this analogy
is more of a mathematical than a physical character.

In Section 3 we clarify the relation between tangles and states of
quantum gravity on $S^3$.  First, the space $\H$ is a
direct sum of subspaces $\H_n$, where $\H_n$ is spanned by tangles with
$2n$ points on the boundary of $D^3$.  
The space spanned by isotopy classes of links in
$S^3$ may be identified with the space $\H_0$.  Second, 
given vectors $\psi,\phi \in \H_n$ corresponding to tangles with
$2n$ boundary points, 
and thinking of these tangles as embedded in two copies
of $D^3$, we may sew the copies of $D^3$ together to obtain a link
in $S^3$, hence a vector $\psi\star \phi \in \H_0$.
In section 3 we use this to construct inner products, or more precisely
nonnegative pairings, on $\H$ from states of
quantum gravity on $S^3$ corresponding to ``nonnegative''
link invariants $\L$.   The pairing of $\psi,\phi \in \H_n$
corresponding to isotopy classes of tangles is given simply by
\[ \langle \psi,\phi\rangle = \L(\psi \star \phi),\]
up to a normalization constant that will prove useful.
Such a pairing typically has vectors with zero norm, but we may
take the quotient of $\H$ by the subspace $\I$ of zero-norm vectors, and
then form the Hilbert space completion $\K$ of $\H/\I$.  Just as the
space of link invariants, which is dual to the space spanned by links,
 may be taken as a state space for quantum gravity on $S^3$,  
we may heuristically regard the dual $\H^\ast$ as a space of states for
quantum gravity on $D^3$.   The process of taking a quotient of $\H$ 
corresponds to taking a subspace of $\H^\ast$, namely
\[  \I^\perp =  \lbrace f \in \H^\ast \colon \; \forall \psi \in \I \;
\; f(\psi) = 0 \rbrace  .\]
However, the inner product on
the quotient $\K$ allows us to identify it with $\I^\perp$.

The significance of this construction may be illuminated
by an analogy.  Consider a quantum system
consisting of two subsystems, so that its Hilbert space may be written
as a tensor product $\H_1 \tensor \H_2$.   
If the whole system is in a pure state $\Psi$, each subsystem, taken
alone, will in general appear to be in a mixed state.  
There is a unique smallest Hilbert space $\K_i 
\subseteq \H_i$ such that the mixed state of the $i$th subsystem
is a mixture of pure states in $\K_i$.   We can think of $\K_i$
as the space of ``partial states'' that arise
by restricting the state $\Psi$ to the $i$th subsystem.     
We may construct $\K_i$ in a
manner very similar to how $\K$ was constructed above.  Namely, we define
a pairing between $\K_1$ and $\K_2$ by
\[      (\psi,\phi) = \langle \Psi,\psi \tensor \phi \rangle .\]
Let $\I_i$ be the subspace of $\K_i$ on which this pairing
vanishes.  Then $\K_i = \I_i^\perp$.   

In the case of quantum gravity, the operation $\psi \tensor \phi$ is
replaced by $\psi \star \phi$, and all need for an inner product has been
avoided through the careful use of duals, but otherwise
things are quite similar.  Thus, if we divide $S^3$ into two halves -
copies of $D^3$ - we may
heuristically regard $\K$ as the space of ``partial states''
of quantum gravity on one half arising by restriction
from the state $\L$ on $S^3$.   Note that by the
diffeomorphism-invariance of the state $\L$, the space $\K$ 
is the same for both halves, and independent of the
particular splitting of $S^3$ into halves.  

The most novel feature is that 
certain states $\L$ corresponding to nonnegative link invariants
determine not only a space $\K$ but also an
inner product on this space.   We show that this inner product gives
a unitary representation of the framed braid group on $\K$, 
consistent with its role as a
symmetry group.   Indeed, in the simplest interesting case, when 
$\L$ is a link invariant known as the Kauffman bracket, this inner 
product is
essentially the only one for which the framed braid group action is
unitary.  

Nonnegative link invariants are also physically interesting since
they may be constructed from Lie group representations as the vacuum
expectation values of Wilson loops in Chern-Simons theory, as shown
by Witten \cite{Witten}.  In Section 4 we recall an
alternate, more rigorous, construction in terms of quantum groups,
due to Reshetikhin and Turaev \cite{Turaev,RT}.  In these 
terms, nonnegative link invariants arise when the parameter $q$ equals
certain roots of unity, these being precisely the values relevant in
rational conformal field theories.  
The Kauffman bracket arises from the spin-${1\over 2}$
representation of $SL(2,\C)$, and is thus a natural
state in the connection representation of quantum gravity as well as
the loop representation.  This state has been examined by
Br\"ugmann, Gambini, and Pullin \cite{BGP}.  We treat this case in
detail.  
In general, for link invariants arising from quantum
group representations we give a simple description of the
$\ast$-algebra structure of the tangle algebra as represented on the
Hilbert space $\K$.   

The author wishes to thank Dan Asimov, Dror Bar-Natan,
Greg Kuperberg, Curtis McMullen,
Geoffrey Mess, Jorge Pullin, J\'ozef Przytycki,
and Stephen Sawin for helpful discussions while preparing the
first version of this paper.  He would also like to thank the
University of Syracuse for its hospitality and a chance to present
this material.  Conversations with Abhay Ashtekar, Bernd Br\"ugmann,
Louis Crane, Renate Loll and Lee Smolin led to a greatly improved
final version of this paper.   

\section{The Tangle Algebra}

We define a {\it tangle} in a manifold with boundary $M$ 
to be a smooth
$1$-dimensional submanifold $X$ of $M$, possibly with boundary, such that 
$\partial X\subset \partial M$ and such that $X$ meets $\partial M$ 
transversally.    
A tangle $X$ is thus a disjoint union of connected components, which
are circles contained in the interior of $M$ or 
arcs connecting two points on $\partial M$.  
The connected components of a tangle will be called {\it strands}, and 
we call the points in $\partial X$ {\it boundary points}. 
There are always an even number of boundary points; if there
are $2n$ boundary points we call $n$ the {\it boundary number} of the
tangle.    

Now suppose that $M$ is a 3-manifold.  
A {\it framing} of a tangle $X \subset M$ is a smooth section $v$ of the
tangent bundle $TM$ over $X$ such that for all $x \in X$, $v_x \notin
T_xX$; for boundary points $x \in X$ we require that $v_x$ is tangent
to $\partial M$.  A framed tangle may be visualized as a disjoint union of
ribbons.  Diffeomorphisms of $M$ act on framed tangles in $M$ in a
natural way.
 We say two framed tangles $(X,v)$ and $(X',v')$ are {\it isotopic} if there is
a continuous one-parameter family $f_t$ of diffeomorphisms of $M$ such
that $f_0$ is the identity, $X' = f_1(X)$, $v' = df_1(v)$, and 
$f_t$ is the identity on $\partial M$ for all $t$.

Often knot theory considers tangles in $[0,1] \times \R^2$.
For us, tangles will be taken in the ball $D^3$ unless otherwise
specified. Let $\widetilde \H$ denote the vector space having as its
basis isotopy classes 
of framed tangles.  Note that
\[   \widetilde \H = \bigoplus_{n=0}^\infty \widetilde \H_n   \]
where $\widetilde \H_n$ is spanned by isotopy classes
of framed tangles with boundary number $n$.    We may further
decompose the spaces $\widetilde \H_n$ as follows.
For any tangle with boundary number $n$,
we write the boundary points as $x = (x_1, \dots, x_{2n})$.
We have $x \in (S^2)^{2n} - \Delta$, where $\Delta$ is the set of 
$2n$-tuples of points in $S^2$ at least two of which are equal.
To each framed tangle we may associate the pair
$(x,v)$, where $x \in (S^2)^{2n} - \Delta$ is as above, and
\[  v = (v_{x_1}, \dots, v_{x_{2n}}) .\]
However, this association is not canonical, since there is no preferred
ordering of the boundary points.
Let ${\cal C}(n)$ denote the space of
pairs $(x,v)$ such that $x \in (S^2)^{2n} - \Delta$
and $v = (v_1, \dots, v_{2n})$, where $v_i$ is
a nonzero vector in $T_{x_i}S^2$.   Let $S_{2n}$ denote the symmetric
group on $2n$ letters.
A permutation $\sigma \in S_{2n}$ acts on
$(x,v) \in {\cal C}(n)$ by permuting the boundary points 
$x_i$ and also the tangent vectors $v_i$:
\[   \sigma(x,v) = (x_{\sigma^{-1}(1)}, \dots, x_{\sigma^{-1}(2n)}, 
v_{\sigma^{-1}(1)}, \dots, v_{\sigma^{-1}(2n)}) .\]
Thus to each framed tangle we may canonically associate {\it boundary
data} $[x,v] \in \B(n)$, where $\B(n)$ is the quotient space ${\cal
C}(n)/S_{2n}$.  Each space $\widetilde \H_n$ is thus a direct sum 
\[   \widetilde \H_n = \bigoplus_{[x,v] \in \B(n)}   \H_n[x,v]   ,\]
where $\H_n[x,v]$ denotes the space spanned by isotopy classes of
framed tangles with boundary data $[x,v] \in \B(n)$.  
The reader may find this uncountable direct sum surprising. 
It is, of course, required by the fact that $\H$ has an uncountable
basis, while each $\H_n[x,v]$ has a countable basis.  Given a measure
on $S^2$, we could replace this direct sum by a direct integral.  
Here, however, we will take advantage of
the fact that the spaces $\H_n[x,v]$ are the fibers of a flat vector
bundle over $\B(n)$.

To see this, first note that each diffeomorphism of $S^2$ extends to a
diffeomorphism of $D^3$, unique up to isotopy.  
This intuitive but highly nontrivial result is due to Cerf, Munkres,
and Smale \cite{Cerf,Munkres,Smale}.  It follows that
$\Diff^+(S^2)$, the group of orientation-preserving
diffeomorphisms of $S^2$, acts on isotopy classes of framed
tangles.  There is thus a representation $\rho$ of $\Diff^+(S^2)$ on
$\widetilde\H$.  Note that if $g \in \Diff^+(S^2)$, 
\[           \rho(g) \maps \H_n[x,v] \to \H_n(g[x,v]) \]
where we define
\[    g[x,v] = [g(x_1), \dots, g(x_{2n}), dg(v_1), \dots, dg(v_{2n})] .\]
It follows that for any $[x,v], [x',v'] \in \B(n)$, we may identify
the spaces $\H_n[x,v]$ and $\H_n[x',v']$.  This
identification is not unique, since there are many $g \in \Diff^+(S^2)$
with $g[x,v] = [x',v']$.   However, if $[x,v]$ and $[x',v']$ are close,
we may take any $g \in \Diff^+(S^2)$ sufficiently close to the
identity with $g[x,v] = [x',v']$, 
and identify $\H_n[x,v]$ with $\H_n[x',v']$ via $\rho(g)$.
The map $\rho(g)$ does not depend on the choice of such $g$, so the
spaces $\H_n[x,v]$ are the fibers of a flat vector bundle over
$\B(n)$.  

This device allows us to reduce the study of the
space $\widetilde \H$ to the study of the spaces $\H_n[x,v]$,
where for each $n \ge 0$, $[x,v]$ is a single arbitrary element of
$\B(n)$.  Thus we fix once and for all distinct points $x_i$ on $S^2$
and nonzero tangent vectors $v_i \in T_{x_i}S^2$ for
$i = 1,2,3,\dots$, and let
\[       \H_n = \H_n[x,v] \]
where $x = (x_1, \dots, x_{2n})$ and
$v = (v_1, \dots, v_{2n})$.    
We let 
\[      \H = \bigoplus_{n=0}^\infty \H_n  .\]
We will often find it convenient to write
\[          x_i^- = x_{2i-1} , \quad x_i^+ = x_{2i}  ,\]
for $1 \le i \le n$, 
and think of the $x_i^-$ as lined up from left to right
 near the north pole of
$S^2$, and the $x_i^+$ similarly lined up near the south pole, with
the corresponding tangent vectors pointing to the right, as in Figure 1.
The boundary points $x_i^-$ will be
called {\it incoming}, while the boundary points $x_i^+$ will be
called {\it outgoing}.
Of course, the actual geometry of the situation is irrelevant, as any
choice of points may be brought into this position by a diffeomorphism.
In a theory based on oriented tangles there would be a real
distinction between incoming and outgoing boundary points; here these
terms are just an arbitary convention.

By forming the space $\H_n$, we have almost reduced the space
$\widetilde \H_n$ by the action of all orientation-preserving
diffeomorphisms of $D^3$.   However, diffeomorphisms which fix the
equivalence class $[x,v]$ still act as
symmetries of $ \H_n$.   Indeed, even a diffeomorphism which fixes
$(x,v)$ can act nontrivially on $\H_n$.  Since the space
$\H_n[x,v]$ is the fiber of a flat vector bundle over $\B(n)$,  the
{\it framed braid group} of $S^2$,
\[       FB_{2n}(S^2) = \pi_1(\B(n)),  \]
acts on $\H_n$ by holonomy.  An explicit presentation of
$FB_{2n}(S^2)$ is known \cite{Baez}.  It has generators
$s_1, \dots, s_{2n-1}$ and $t_1, \dots, t_{2n}$, and relations
\ba   s_j s_k &=& s_k s_j   
\qquad {\rm if}\qquad  |j - k| \ge 2, \nonumber\cr
s_j s_{j+1} s_j &=& s_{j+1} s_j s_{j+1} \nonumber\cr
s_j t_k &=& t_k s_j  \qquad{\rm if} \qquad    k \ne j, j+1 , \nonumber\cr
t_{j+1}s_j &=& s_j t_j , \nonumber\cr
t_j s_j &=& s_j t_{j+1}, \nonumber\cr
  s_1s_2 \dots s_{n-1}s_{n-1}s_{n-2} \dots s_1 &=& t_1^2 .
\ea
For $1 \le j < n$,
the generator $s_j$ corresponds to switching 
the incoming boundary points $x_j^-$
and $x_{j+1}^-$ in a counterclockwise fashion, as
in Figure 2.   Similarly, for $n < j \le 2n-1$, $s_j$ switches
the outgoing points $x_{2n-j}^+$ and $x_{2n-j+1}^+$.
The generator $s_n$ switches the incoming point $x_n^-$ with the
outgoing point $x_n^+$.   
For $1 \le j \le n$, the generator $t_j$ adds a counterclockwise
$2\pi$ twist to the framing at
$x^-_j$, while for $n < j \le 2n$, $t_j$ adds a twist
to the framing at $x^+_{2n-j}$, as in Figure 3.
We will also regard the elements $g \in FB_{2n}(S^2)$ as
operators on $\H$ by letting them act as zero on all
$\H_m$ with $m \ne n$.

An element of $FB_{2n}(S^2)$ may be represented as a loop in 
the space $\B(n)$, or as a path in ${\cal C}(n)$.  It may thus be
represented as a particular sort of framed tangle
in $[0,1] \times S^2$ with boundary number $2n$, having the points
$(0,x_i)$ and $(1,x_i)$ as boundary points.
The framing at the boundary points must match up with the
standard vectors $v_i$ in the obvious way.  Moreover, the tangle
cannot contain any embedded circles, and all the embedded line
segments must be of the form
\[          t \mapsto (t,f(t)) \]
for some $f \maps [0,1] \to S^2$.  In these terms, the action of  
of $FB_{2n}(S^2)$ on $\H_n$ corresponds to sewing a tangle in
$[0,1] \times S^2$ onto a tangle in $D^3$ to obtain a tangle in a larger
copy of $D^3$, as in Figure 4.  

The natural symmetry group of $\widetilde \H$ is $\Diff^+(S^2)$. 
Thus it is natural to seek inner products on $\widetilde \H$ for which
the representation $\rho$ of $\Diff^+(S^2)$ becomes unitary.  
For the reduced space $\H$, diffeomorphism-invariance is encoded
in the representations of the framed braid
groups $FB_{2n}(S^2)$ on each summand $\H_n$.  It is easy to check that
from any choice of
inner products on $\H_n$ for which the framed braid group representations
are unitary we obtain an inner product on $\widetilde \H$ for which
$\rho$ is unitary, via the isomorphism $\H_n[x,v] \simeq \H_n$.  Conversely,
any inner product on $\widetilde \H$ for which the action of
$\rho$ is unitary and for which the subspaces $\H_n[x,v]$ are
orthogonal determines inner products on the $\H_n$ for
which the action of $FB_{2n}(S^2)$ is unitary.  This is the precise 
sense in which the reduced space $\H$ allows us to study $\widetilde \H$.
In fact, in the next section we obtain pairings on the $\H_n$ that are
preserved by the framed braid group representation, but these pairings
have zero-norm states.  This gives us a diffeomorphism-invariant inner
product not on $\widetilde\H$, but on a quotient thereof.   

We now define some other operators on $\H$ which give a useful
computational framework in which to do ``tangle field theory.'' 
First, note that framed tangles with boundary number zero may be
identified with framed links in $D^3$, or equivalently, in $S^3$.  Thus
$\H_0$ has as a basis the isotopy classes of framed links in $S^3$.
Let the {\it empty link}, $\psi_0 \in \H_0$, be the basis vector
corresponding to empty set, which is vacuously a framed link.
The vector 
$\psi_0$ is roughly analogous to the vacuum in quantum 
field theory, in that all
vectors in $\H$ may be obtained from $\psi_0$
by applying certain operators and taking linear combinations, as we
now describe.  

The {\it creation} operator 
\[    c \maps \H_n \to \H_{n+1}   \]
simply adds an extra strand to any tangle with boundary number $n$;
the new strand is required to be unknotted, untwisted,
and to remain to the right
of the existing strands.  Figure 5 shows a tangle representing
 $\psi \in \H_n$ and the tangle representing $c\psi$.   For
$n \ge 1$, the {\it annihilation} operator
\[   a \maps \H_n \to \H_{n-1}  \]
moves the rightmost boundary points, $x_n^-$ and $x_n^+$, slightly
into the interior of $D^3$, and connects them with a smooth arc that
is unknotted, untwisted, 
and remains to the right of the existing strands.   We
define $a$ to be zero on $\H_0$.  Figure 6 shows a tangle representing
$\psi \in \H_n$ and the tangle representing $a \psi$.
It is essential, but easy, to check that the annihilation and creation
operators are well-defined.

The importance of the creation and annihilation operators is that, 
together with the operators
coming from the framed braid groups, they generate all operations
on tangles which can be performed by manipulations at the boundary of 
$D^3$: creating new strands, braiding and twisting existing strands
that meet the boundary, and ``closing off'' such strands.   
This can be made completely precise.  Define the
{\it tangle algebra} $\T$ to be the
algebra of operators on $\H$ generated by the operators
$ap_n$, $cp_n$, and $g$ for all $n \ge 0$ and $g \in FB_{2n}(S^2)$, 
where $p_n \maps \H \to \H_n$ is the projection.  (For technical
reasons, this is more convenient than the algebra generated by
$a$, $c$, and $g \in FB_{2n}(S^2)$.)  
We may represent a product of such operators as a
tangle in $[0,1] \times S^2$ having as boundary points the points
$(0,x_1), \dots , (0,x_n)$ and $(1,x_1), \dots, (1,x_m)$ for some
$n,m \ge 0$.  The product in the tangle algebra corresponds
to sewing tangles in two copies of $[0,1] \times S^2$ along one boundary
to obtain a tangle in a thicker copy of $[0,1] \times S^2$, as
in Figure 7.   Just as with the action of the framed braid group,
the action of the tangle algebra on $\H$ corresponds to sewing tangles in
$[0,1] \times S^2$ onto tangles in $D^3$ to obtain tangles in
$D^3$.   We show in the Appendix that every vector in $\H$ is of the form
$T\psi_0$ for some $T \in \T$, i.e., the empty link
is a cyclic vector for the tangle algebra. 


\section{Tangle Field Theories}  

By a {\it link invariant} we mean a complex-valued invariant of framed
unoriented links in $S^3$, or equivalently, a linear functional
$\L \maps \H_0 \to \C$.  
Here we show that certain ``nonnegative'' link invariants
define inner products on quotients of $\H$. 

First, note that since two copies of $D^3$ may be sewn together along
a 2-disc to obtain another copy of $D^3$, each space $\H_n$ becomes
an algebra, with the product $\psi \phi$ of two framed 
tangles $\psi$ and $\phi$ given by attaching
each boundary point $x_i^+$ of $\psi$ to the 
boundary point $x_i^-$ of $\phi$, as in Figure 8.  
The identity of $\H_n$ is the vector $ c^n \psi_0$.  
This remarkable fact, that the space $\H_n$
has an algebra structure, also holds for the state
space of $S^n$ in any topological quantum field theory satisfying
Atiyah's axioms \cite{Atiyah}.
Moreover, $\H_n$ becomes a $\ast$-algebra in a unique manner such that 
for $\psi\in \H_n$ the isotopy class of a tangle, $\psi^\ast$ is the
the isotopy class of the tangle given by reflecting $\psi$
about the $xy$-plane as in Figure 9.   
Here we assume, without loss of generality, 
that reflection of the points
$x_i^-$ and vectors $v_i^-$ about the $xy$-plane yields
$x_i^+$ and $v_i^+$, respectively.  

Next, given a link invariant $\L$ and a complex number $N$, 
we may define a trace on each algebra $\H_n$ as follows:
\[     \tr(\psi) = N^n \L(a^n\psi) \]
for $\psi \in \H_n$.  Here we identify $a^n\psi$, the {\it closure} of
the framed tangle $\psi$, with a framed link in $S^3$.  See Figure 10
for a picture of $a^n\psi$.  It is easy to check that 
$\tr\colon \H_n \to \C$ 
is indeed a trace, that is, a linear functional with $\tr(\psi\phi) = 
\tr(\phi\psi)$.    

It is convenient to make all of $\H$ into a $\ast$-algebra as the
direct sum of the $\ast$-algebras $\H_n$, so that the product of 
$\psi \in \H_n$ and $\phi \in \H_m$ vanishes by convention if $n \ne
m$.  We may then define a trace $\tr \maps \H \to \C$ whose
restriction to each summand $\H_n$ is given as above.
Given a link invariant $\L$, we may thus define a pairing
on $\H$ by setting
\[  \langle \psi,\phi \rangle  = \tr(\psi^\ast\phi) .\]
This is not generally an
inner product, but it is linear in the second argument and
conjugate-linear in the first, and satisfies
\[        \langle \psi,\phi \rangle = \langle \phi^\ast,\psi^\ast\rangle,\]
\[ \langle \psi\phi,\eta \rangle = \langle \phi, \psi^\ast\eta \rangle .\]
Moreover, on each summand $\H_n$ this pairing is automatically
preserved by the action of the framed braid group, since
for all $g \in FB_{2n}(S^2)$ and $\psi,\phi \in \H_n$,
\ba    \langle g\psi,g\phi \rangle &=&
N^n \L\left(a^n((g\psi)^\ast (g\phi))\right)  \nonumber\cr
&=& N^n \L\left(a^n (\psi^\ast\phi)\right) \nonumber\cr
&=& \langle \psi,\phi\rangle. \nonumber\ea
Here we use the fact that
\[ a^n((g\psi)^\ast (g\phi)) = a^n(\psi^\ast\phi) .\]
   We also have useful identities
involving the annihilation and creation operators.  If $\psi \in \H_{n-1}$
and $\phi \in \H_n$, 
\ba      \langle c\psi,\phi\rangle &=&  N^n \L(a^n ((c\psi)^\ast
\phi))) \nonumber\cr
&=&  N^n \L(a^{n-1} (\psi^\ast (a\phi))) \nonumber\cr
&=&  N \langle \psi,a\phi\rangle \nonumber\ea
where we use the fact that
\[   a^n ((c\psi)^\ast \phi)) =  a^{n-1} (\psi^\ast (a\phi)) .\]
 It follows that for all $\psi,\phi \in \H$, 
\[      \langle c\psi,\phi \rangle = N \langle \psi,a\phi \rangle .\]
Similarly,
\[      \langle \psi, c\phi \rangle = N\langle a\psi, \phi \rangle .\]

A more invariant definition of the pairing associated to a link
invariant is as follows.  Given $\psi,\phi \in \H_n$ corresponding to
framed tangles in two copies of $D^3$, we may sew together the
copies of $D^3$ by an orientation-reversing map along the boundary and obtain a
framed link $\psi\star\phi$ in $S^3$.  Then 
\[        \langle \psi,\phi\rangle = N^n \L(\psi\star\phi) .\]
It is worth emphasizing that the product in the tangle algebra, the action
of the tangle algebra on $\H$, the product on each $\H_n$, and the
pairing on $\H_n$ all arise from sewing.  

In general the pairing associated to a link invariant will be
degenerate, that is, there will be nonzero vectors lying in
\[    \I_n = \lbrace \psi \in \H_n \colon\; \forall \phi \in \H_n\;\;
\langle \psi, \phi\rangle= 0 \rbrace.\]
We define 
\[    \I = \bigoplus_{n=0}^\infty \I_n .  \]
As discussed in the introduction, we may think of the quotient
$\H/\I$ as a sort of
 space of ``partial states'' for quantum gravity on $D^3$ associated
to the state $\L$.

Now suppose that the link invariant $\L$ is {\it nonnegative,} that is, 
for some choice of $N \ne 0$, $ \langle \psi,\psi \rangle \ge 0$ 
for all $\psi \in \H$.  This requirement determines the choice of $N$
up to a positive factor.  It follows that
\[        \langle \psi, \phi\rangle = \overline{\langle\phi,
\psi\rangle}  .\]
Thus the pairing on $\H_n$ gives rise to a true
inner product on the quotient space $\H_n/\I_n$.  We may complete this
quotient to form a Hilbert space $\K_n$, and define 
$\K$ to be the Hilbert space direct sum
\[      \K = \bigoplus_{n=0}^\infty \K_n.   \]
Alternatively, we may think of $\K$ as the Hilbert space completion of
$\H/\I$.  We call $\K$ the {\it tangle field
theory} associated to the state $\L$.  

The quotient $\H/\I$ inherits the key algebraic structures possessed
by $\H$, namely, it is both a $\ast$-algebra and a representation
of the tangle algebra. 
To see this, first note that $\I$ is a two-sided ideal of $\H$, 
since if $\psi \in \I$, for any $\eta \in \H$ we have
$      \langle\psi\eta,\phi\rangle
 = \langle \psi,\phi\eta^\ast \rangle = 0$ 
and
$    \langle \eta\psi,\phi\rangle = \langle \psi,\eta^\ast\phi \rangle
= 0$
for all $\phi \in \H$.
Also, $\I$ is preserved by the $\ast$ operation on $\H$, since if
$\psi \in \I$, 
$    \langle \psi^\ast,\phi\rangle = \langle \phi^\ast,\psi\rangle
= 0 $
for all $\phi \in \H$, so $\psi^\ast \in \I$.  It follows that 
the quotient $\H/\I$ inherits the structure of a $\ast$-algebra
from $\H$.  Next, note that if $\psi \in \I$, for all $\phi \in \H$ we have
$   \langle a\psi,\phi \rangle = N^{-1} \langle \psi, c\phi\rangle =
0$ and $    \langle c\psi, \phi\rangle = N\langle \psi, a\phi\rangle = 0$,
and if in addition $\psi \in \I_n$ and $g \in FB_{2n}(S^2)$, 
$   \langle g\psi, \phi\rangle = 
\langle \psi, g^{-1}\phi\rangle = 0$.  
It follows that $\I$ is preserved by the action of the tangle algebra
$\T$, so $\T$ acts on $\H/\I$.     

The next step is to extend these algebraic structures from $\H/\I$ to
the Hilbert space completion $\K$.  It follows from the results above 
that for each $n$, the representation of the framed
braid group $FB_{2n}(S^2)$ on $\H_n/\I_n$ extends to a
unitary representation on the Hilbert space $\K_n$.   This is
consistent with the notion 
that the framed braid group acts as symmetries of $\K_n$.

To show that the representation of the tangle algebra on $\H/\I$
extends to a representation on $\K$, it suffices to prove the bound
$ \|c\psi\| \le K \|\psi\| $ 
for some constant $K$, as a similar bound for the annihilation operator
then follows from the relation $\langle c\psi,\phi\rangle = 
N \langle \psi,a\phi\rangle$.   In the next section we prove this for
link invariants constructed from quantum group representations.

To extend the $\ast$-algebra structure from $\H/\I$ to $\K$ also requires
proving a bound.  It is easy to see that the $\ast$ operation on
$\H/\I$ satisfies 
\[      \|\psi^\ast\| = \|\psi\| ,\]
hence extends by continuity to $\K$.  If one can show that
$ \|\psi\phi \| \le \|\psi\| \|\phi\| $, 
the product also extends by continuity to $\K$, and $\K$ becomes what
is known as an ``H*-algebra'' \cite{Ambrose}.  The
structure theory of H*-algebras is completely understood; for example,
every finite-dimensional H*-algebra is a direct sum of matrix
algebras.  In the examples in the next section, arising from
quantum group representations at roots of unity,
each summand $\K_n$ is a finite-dimensional H*-algebra.   
It is easy to see that in this case the action of the
tangle algebra extends by continuity from $\H/\I$ to $\K$, so that
$\K$ inherits all the key algebraic structures of $\H$. 
 
If each $\K_n$ is a finite-dimensional H*-algebra,
we could call $\K$ a {\it rational} tangle field theory.
These are closely tied to rational
conformal field theories, as should be clear from the following
section together with the work on 2-dimensional physics and
3-dimensional topology by Witten \cite{Witten}, Crane \cite{Crane},
Fr\"ohlich and King \cite{FK}, and others.  In part, this connection is 
due to the fact that the representation of the framed braid group on
$\H_n$ 
factors through a quotient, the {\it framed mapping class group}, in which
the relation
\[       (s_1 \cdots s_{2n-1})^{2n} = 1 \]
holds.  These groups are extensions of the usual mapping class groups
of the sphere \cite{Birman}.    

\section{Tangle Field Theories from Quantum Groups}

A very important construction of link invariants using quantum groups
was given by Reshetikhin and Turaev \cite{Turaev,RT}.  Given any
finite-dimensional irreducible representation $V$ of a simple Lie group,
they obtain an invariant of framed oriented links depending on a
complex parameter $q$.    Given an isomorphism
between $V$ and the dual representation $V^\ast$, one in fact obtains
an invariant of framed unoriented links.   

To begin with, consider the simplest example, based on
the spin-${1\over 2}$ representation of $SU(2)$.  This yields a link
invariant known as the Kauffman bracket \cite{Kauffman}. 
(The Jones polynomial of a given
framed oriented link is simply the Kauffman bracket of the
corresponding unoriented framed link times
$(-q)^{3w/4}$, where $w$ is the writhe of the link.  The Jones
polynomial turns out to be independent of the framing.)  
The Kauffman bracket is usually normalized so that its value on the
unknot is $1$, but we will work with a normalization $\L$ that equals $1$
on the empty link:
\be     \L(\psi_0) = 1.  \label{empty}\ee
This version of the Kauffman bracket, just as the usual one,
may be recursively calculated using the skein relations 
\be      \L(\phi_+) = q^{-1/4} \L(\phi_0)  + q^{1/4} \L(\phi_\infty)  
\label{skein} \ee
where $\phi_+, \phi_0, \phi_\infty \in \H_0$ are the
isotopy classes of three framed links that are identical except
within a small ball, where they appear as in Figure 11, and
\be     \L( ac\psi) = -(q^{1/2} + q^{-1/2}) \L(\psi)
\label{unknot}\nonumber\ee
where $\psi \in \H_0$, and $ac\psi$, as can easily be seen, is the
``distant union'' of $\psi$ with a trivially framed unknot.  

The Kauffman bracket and its relation to the algebra of tangles has
been extensively studied, and we will simply review the implications
for the trace on $\H_n$ and the associated quotient $\H_n/\I_n$. 
Suppose that $q$ is such that $\L$ is nonnegative.
It is easy to see that by repeated use of skein relations (\ref{skein})
and (\ref{unknot}) one can reduce any framed tangle with isotopy class $\psi
\in \H_n$ to a linear combination of braids.  Thus the algebra $\H_n/\I_n$ 
is a quotient of the braid group algebra, ${\C}B_n$.  This 
is the algebra with generators $g_i$ for $g_i^{-1}$ for $1 \le i < n$, 
and relations 
\ba    g_ig_j &=& g_j g_i \qquad\qquad\qquad |i-j| > 1, \nonumber\cr
g_ig_{i+1}g_i &=& g_{i+1}g_i g_{i+1}. \nonumber\ea
In fact \cite{Jones,Kauffman}, one can easily use the skein relations
to show that in $\H_n/\I_n$ the images of the $g_i$ satisfy the relations
\[ g_i^2 = (q-1)g_i + q \]
that define the Hecke algebra as a quotient of the braid group
algebra, and the extra relations 
\[   g_ig_{i+1}g_i + g_ig_{i+1} + g_{i+1}g_i + g_i + g_{i+1} + 1 = 0  \]
defining the Temperley-Lieb algebra as a quotient
of the Hecke algebra.  

In short, $\H_n/\I_n$ is a quotient of the Temperley-Lieb
algebra, so the trace on $\H_n$ (hence $\H_n/\I_n$) defines a trace on 
the Temperley-Lieb algebra.   This trace has been
thoroughly studied by Jones
\cite{Jones2}, and it follows easily from his work that the 
link invariant $\L$ is nonnegative if and only if
\[               q = e^{\pm 2\pi i/\ell} \]
with $\ell \ge 3$ an integer.   Of course, the trace on $\H_n$
depends on a constant $N$, with the condition that $\tr(\psi^\ast\psi)
\ge 0$ 
fixing $N$ up to a positive constant.  We may satisfy this condition by
taking
\[      N^{-1} = -(q^{1/2} + q^{-1/2}) .\]
The advantage of this precise normalization will be explained below.
As shown by Jones, in this situation each $\H_n/\I_n$ is 
finite-dimensional, so $\H_n/\I_n$ is equal to its Hilbert space
completion $\K_n$.  Moreover, each $\K_n$ is an H*-algebra
whose structure is precisely understood.   
To summarize, $\K$ is a rational tangle field theory.

By the general theory of the previous section, the space of states
$\K_n$ is a unitary representation of the framed braid group
$FB_{2n}(S^2)$.  In the Appendix we show that it is actually an
irreducible representation.  This implies that the inner product we
have chosen for $\K_n$ is the only one (up to a constant factor)
preserved by the framed braid group action, exemplifying 
the principle that inner products
should be determined by the constraint that symmetry groups act
unitarily.    

The Kauffman bracket may be defined either by skein relations as
above, or
in terms of the spin-${1\over 2}$ representation of quantum $SU(2)$, or
via Chern-Simons theory with gauge group $SU(2)$.  We have used the skein
relations above because they are simple, but the other two approaches
are more general.  
It should be noted that the Kauffman bracket 
is implicit in the work of Br\"ugmann, Gambini and Pullin \cite{BGP},
who obtain the Jones polynomial times a function of
the writhe when constructing states of quantum gravity from $SU(2)$
Chern-Simons theory.  The above results effectively determine the
inner product on ``Chern-Simons states'' of quantum gravity on $D^3$
from the unitarity of the framed braid group action.   

Now let us turn to what can be said in general about tangle field
theories obtained from quantum group representations.  Let $V$ be a
finite-dimensional representation of a simple Lie group.  Then for
each nonzero $q \in \C$, $V$ may be regarded as a representation of the 
corresponding quantum group.   Suppose that we are given a fixed
isomorphism of $V$ and $V^\ast$ as quantum group representations.  Then
the construction of Reshetikhin and Turaev defines, for each $n \ge 0$, a
homomorphism $\L_n$ from the algebra $\H_n$ to the algebra of 
linear transformations of $V^{\tensor n}$.  
In particular, taking $n = 0$, we obtain a link invariant
$\L_0 \maps \H_0 \to \C$, which we write simply as $\L$.

The link invariants defined this way have many special properties.
In particular, there is always a constant $N^{-1}$ (depending on $q$)
such that 
\[          \L(ac\psi) = N^{-1} \L(\psi)  \]
for all $\psi \in \H_0$.  (We restrict ourselves to the case where
this constant is nonzero, so that it can be written as $N^{-1}$.)  
Recall that if $\psi$ is the isotopy class
of a link, $ac\psi$ is the isotopy class of the distant union of that
link with the trivially framed unknot.  The constant $N^{-1}$ is called
the {\it loop value} of the link invariant $\L$, and we will use this
choice of $N$ in the definition of the pairing on $\H_n$:
\[         \langle \psi,\phi \rangle = N^n \L(\psi^\ast\phi)  .\]

It follows that for all $\psi,\phi \in \H_n$,
\ba      \langle ac\psi,\phi \rangle 
&=& N^n \L\left(a^n((ac\psi)^\ast \phi)\right) \nonumber\cr
&=& N^n \L\left(aca^n(\psi^\ast\phi)\right) \nonumber\cr
&=&   N^{n-1}  \L\left(a^n(\psi^\ast \phi)\right)   \nonumber\cr
&=&   N^{-1} \langle \psi, \phi\rangle , \nonumber\ea
so 
\[                  ac = N^{-1} \]
as an operator on $\H$.  In particular, it follows that the trace 
 of the identity of $\H_n$ is $1$:
\[   \tr(c^n\psi_0) = N^n \L(a^nc^n\psi_0) = N^n \L((ac)^n \psi_0)
= \L(\psi_0) = 1 . \]

The identity $ac = N^{-1}$ has an important consequence
when $\L$ is nonnegative.  In this case, for any $\psi \in
\H/\I$ we have
\[      \|c\psi\|^2 = N\langle \psi,ac\psi \rangle = \|\psi\|^2 . \]
As described at the end of the previous section, this implies that the
representation of the tangle algebra on $\H/\I$ extends to the Hilbert
space completion $\K$.  Moreover, the creation operator acts as an
isometry on $\K$, and 
\[                  c^\ast = Na  .\]
Thus the $\ast$-algebra structure of the tangle algebra as represented
on $\K$ is determined solely by the loop value of $\L$.

In tangle field theories associated to gauge groups other
than $SU(2)$, the HOMFLY polynomial invariant of links and the Hecke
algebra will arise when working with $SU(n)$
\cite{Jones,HOMFLY,Wenzl}, while the Kauffman polynomial and the
Birman-Wenzl algebra will arise when working with $SO(n)$
\cite{Kauffman2,BW,Wenzl2}.  Wenzl and Ocneanu
have done all the work necessary to 
determine which values of $q$ give rise to rational tangle field
theories in these cases.   The work of Yetter \cite{Yetter} and
Turaev \cite{Turaev2}
 is also very useful for understanding the general situation. 


\section{Conclusions}

We have given a construction of Hilbert spaces $\K$ of 
tangles on $D^3$
from states of quantum gravity on $S^3$ that correspond to nonnegative
link invariants.  While this construction is mathematically very
natural, its physical significance still needs to be evaluated.
Most importantly, one would like to understand how
seriously one can take the notion of $\K$ as a space
of ``states of quantum gravity on $D^3$.''   
 Bound up with this is the question of the real physical meaning of 
nonnegativity for states on $S^3$.
So far the only clues come from their intimate
connection with rational conformal field theories
and 3-dimensional topological quantum field theories.
It would be preferable to find an interpretation of nonnegativity
directly in terms of 3+1-dimensional physics, if possible.

For a closer contact between tangle field theories and traditional
physics, however, the most promising approach may involve
asymptotically flat quantum gravity.  
Classically, we may extend the Cauchy data
for asymptotically Minkowskian solutions from $\R^3$ to a $D^3$
completion \cite{Ashtekar2}.    Thus it is natural to attempt to give a
loop variables formulation of quantum gravity in this case in terms of
tangles on $D^3$.  The group of spatial symmetries in this
case, however, is the Poincar\'e group rather than $\Diff^+(S^2)$, so
one does not expect the formalism given here to be applicable without
substantial adaptation.
One would hope to constrain the inner product by requiring that
the Poincar\'e group act unitarily.  This would permit
a connection to traditional observables such as
energy, momentum, and angular momentum.  

It would also be interesting to consider
theories based on framed tangles admitting some sort of
self-intersections.  
This is desirable because only states built from self-intersecting links
are not annihilated by the determinant of the metric.  
Indeed, whether states built from
non-self-intersecting links are ``physical'' is a matter of dispute
\cite{Smolin,ARS}, and there has been considerable work on finding
solutions to the Hamiltonian constraint built from links with
intersections \cite{BP,Husain,JS}.  It is interesting to note
that such states may be constructed by a suitable extension of
the Kauffman bracket to links with intersections \cite{BGP}.
While this topic is still
not fully understood, relevant mathematical techniques for
dealing with links admitting self-intersections have recently
been developed by Vassiliev
\cite{Vassiliev} and, in subsequent work, Bar-Natan \cite{Bar-Natan},
Birman and Lin \cite{BL}, and others.

Finally, it is worth noting that tangle field theories are closely
related to what knot theorists call ``skein modules,'' and that skein
modules suggest generalizations of tangle field theories to 3-manifolds
other than $D^3$.  There is a review article on skein modules
by Hoste and Przytycki \cite{HP}.  Nonnegative link invariants
arising from quantum group representations at roots of unity
can be generalized from $S^3$ to all compact
oriented 3-manifolds, as is clear from the work of Witten, Crane,
 Reshetikhin, Turaev, and others \cite{Witten,Crane,RT2}.


\section{Appendix}

In this Appendix we prove two results mentioned above: that the empty
link is a cyclic vector for the tangle algebra, and that the framed
braid group $FB_{2n}(S^2)$ acts irreducibly on the space $\K_n$
appearing in the tangle field theory associated to the Kauffman bracket.

For the first result we give a purely algebraic proof based on the
presentation by Yetter \cite{Yetter} and Turaev \cite{Turaev2} of the
category of isotopy classes of tangles in $[0,1] \times \R^2$.  
Note that isotopy classes of such tangles are equivalent to isotopy
classes of tangles in $D^3$ as we have been dealing with them.
Let $\Hom(n,m)$ denote the set of isotopy classes of framed tangles in 
$D^3$ having as boundary data 
\[    [x_1^-, \dots, x_n^-, x_1^+, \dots , x_m^+, 
v_1^-, \dots, v_n^-, v_1^+, \dots , v_m^+]   .\]
We use this notation since isotopy classes of framed tangles may be
regarded as the morphisms of a category $FT$ with objects labelled by
the natural  numbers $\lbrace 0,1,2,\dots\rbrace$.  Given $\phi \in
\Hom(n,m)$ and $\psi \in \Hom(m,k)$, the composite $\psi\phi \in
\Hom(n,k)$ is defined in the obvious way, as in Figure 8.  This category
is a strict monoidal category in which the tensor product of objects is
defined by $n \tensor m = n+m$, and the tensor product $\psi \tensor
\phi \in  \Hom(n+n',m+m')$ of $\psi \in \Hom(n,m)$ and $\phi \in
\Hom(n',m')$ is given by setting $\psi$ and $\phi$ side to side as in
Figure 12. We define the elements $| \in \Hom(1,1)$,
$r, r^{-1} \in \Hom(2,2)$, $\cup \in \Hom(2,0)$, $\cap \in \Hom(0,2)$ as 
in Figure 13.   Note that $| \in \Hom(1,1)$ is the identity morphism.

Yetter and Turaev showed that the 
category $FT$ is generated 
by the morphisms $|,r,r^{-1},\cup,$ and $\cap$ by taking composites and
tensor products.  In fact, they gave a presentation of $FT$ as a monoidal
category with these generators together with the relations
\ba   rr^{-1} &=& r^{-1}r \quad =\quad | \tensor | \nonumber\cr
  (| \tensor \cup)(\cap \tensor |) &=& (\cup \tensor |)(| \tensor
\cap) \quad =\quad |  \nonumber\cr
  (r \tensor |)(| \tensor r)(r\tensor |) &=& (| \tensor r)(r \tensor
|)(1 \tensor r) \nonumber\cr
(| \tensor \cup)(r \tensor |) &=& (\cup \tensor |)(| \tensor r^{-1}) 
\nonumber\cr 
(| \tensor \cup)(r^{-1} \tensor |) &=& (\cup \tensor |)(| \tensor r)
\nonumber\cr 
(r \tensor |)(| \tensor \cap) &=& (| \tensor r^{-1})(\cap \tensor |)
\nonumber\cr
(r^{-1} \tensor |)(| \tensor \cap) &=& (| \tensor r)(\cap \tensor |)
\nonumber\cr
(| \tensor \cup)(r \tensor |)(| \tensor \cap) &=& (\cup \tensor |)
(| \tensor r^{-1})(\cap \tensor |) .\nonumber\ea
The reader is urged to draw the tangles corresponding these
identities and verify them.  

Note that $\H_n$ is just the vector space having as basis the elements of 
$\Hom(n,n)$.   The representation of $FB_{2n}(S^2)$ on $\H_n$ comes from an
action as permutations of this basis.  One can describe this action
explicitly.  Let $|^n \in
\Hom(n,n)$ denote the $n$-fold tensor product of the morphism $|$;
$|^n$ is the identity morphism in $\Hom(n,n)$.  Define
\ba    r_i &=& |^{i-1} \tensor r \tensor |^{n-i-1} \in \Hom(n,n), \nonumber\cr
    \cup_i &=& |^{i-1} \tensor \cup \tensor |^{n-i-1} \in \Hom(n,n-2), 
\nonumber\cr
    \cap_i &=& |^{i-1} \tensor \cap \tensor |^{n-i+1} \in \Hom(n,n+2). 
 \nonumber\ea
Then the action of $FB_{2n}(S^2)$ on $\Hom(n,n)$ is given by
\be    s_i\psi = \left\{ \begin{array}{ll}
\psi r_i         & 1 \le i < n        \\
\cup_n (\psi \tensor r^{-1}) \cap_n         & i = n	\\
r_{2n-i}^{-1} \psi          & n < i \le 2n-1 
\end{array} \right.        
 \label{braid} \ee
and
\be    t_i\psi = \left\{ \begin{array}{ll}
\psi \cup_{i+1} r_i \cap_{i+1}         & 1 \le i \le n        \\
 \cap_{2n-i+1} r_{2n-i+2} \cup_{2n-i+1} \psi   & n < i \le 2n	
\end{array} \right.        
 \label{twist} \ee
Moreover, for $\psi \in \Hom(n,n)$ we have
\ba      c\psi &=& \psi \tensor | , \cr
       a\psi &=& \cap_n(\psi \tensor |)\cup_n .\label{annihilate} \ea

\begin{theorem}\DOT  The vector $\psi_0$ is a cyclic vector for the
representation of $\T$ on $\H$.
\end{theorem}

Proof - First note that there is a one-to-one and onto function $S \maps
\Hom(p,q) \to \Hom(p+1,q-1)$ if $q \ge 1$, given by
\[        S\psi = \cup_q(\psi \tensor |) ,\]
with inverse given by
\[        S^{-1}\psi = (\psi \tensor |)\cap_{p+1}.\]
By composing these functions we get one-to-one and onto functions
\[        S^k \maps \Hom(p,q) \to \Hom(p+k,q-k)  \]
for any integer $k$ such that $p+k,q-k > 0$.

We may use these functions to express the basic tangles $r_i, 
r_i^{-1}, \cup_i,$ and $\cap_i$ in terms of elements of the tangle
algebra, as follows.
Suppose that $\psi \in \Hom(p,q)$, let $(q-p)/2 = k$, which must
be an integer if $\Hom(p,q)$ is nonempty, and let $n = (p+q)/2$.
Then the following facts are consequences of
(\ref{braid}-\ref{annihilate}), together with the Yetter-Turaev relations.
(Actually, these facts are most easily seen using pictures.)
For all $1 \le i < q$, 
\[     r_i\psi  =         S^{-k}gS^k\psi \]
for some $g \in FB_{2n}(S^2)$, and similarly for $r_i^{-1}\psi$.  
Also, for all $1 \le i < q$, 
\[       \cup_i \psi =  S^{-k+1}ag S^k\psi \]
for some $g \in FB_{2n}(S^2)$, and for $1 \le i \le q+1$,
\[       \cap_i \psi = S^{-k-1}gcS^k \psi  \]
for some $g \in FB_{2(n+1)}(S^2)$.   

By the result of Yetter and Turaev, for all $n,m$, 
every element of $\Hom(n,m)$ is a product of elements of the form
$r_i, r_i^{-1} \in \Hom(p,p), \cup_i \in \Hom(p,p-2),$ and $\cap_i
\in \Hom(p,p+2)$, where $p$ is arbitrary.  Suppose that
we write $\psi \in \Hom(n,n)$ as a product
\[   \psi = x_k x_{k-1} \cdots x_1 \]
of this form, and suppose $x_1 \in \Hom(p,q)$.  
Then 
\[   \psi = x_k \cdots x_1 c^p \psi_0 .\]
By the facts in the previous paragraph, we can rewrite the expression
on the right hand as $T\psi$, where $T$ lies in the tangle algebra
$\T$.  (Note that all the powers of $S$ cancel.)  
\qed

Our second result concerns the tangle field theory $\K$ associated to
the Kauffman bracket.  Recall from the previous section that when $q =
\exp(\pm 2\pi i/\ell)$ with $\ell \ge 3$, each summand $\K_n$ is a finite
direct sum of matrix algebras.  

\begin{theorem}\DOT  Suppose that $q = \exp(\pm 2\pi i/\ell)$ with $\ell \ge
3$.  Then the representation of $FB_{2n}(S^2)$ on $\K_n$ is
irreducible.  \end{theorem}

Proof - Given the isotopy class of any
framed tangle in $\Hom(n,n)$, the relations in the
Temperley-Lieb algebra may be used to rewrite it as a linear
combination of products of elements of the form $\cup_k\cap_k \in
\Hom(n,n)$, as noted by Kauffman \cite{Kauffman}.    Thus $\K_n$ is
spanned by such products.   Given such a product, we say that ``$i$
strands go through'' if $i$ strands connect boundary points at the
top, $x_j^-$, to boundary points at the bottom, $x_j^+$.   If $n$ is
even (resp.\ odd), an even (resp.\ odd) number of strands must go
through.   If $n$ is even, let $J_i$ denote the subspace of $\K_n$
spanned by 
products for which $\le 2i$ strands go through.    If $n$ is odd, let
$J_i$ be the subspace spanned by products for which $\le 2i + 1$ strands
go through.  Here it suffices to consider $0 \le i \le \lfloor n/2
\rfloor$. 
Note that each space $J_i$ is a two-sided ideal, 
and $J_0 \subseteq J_1 \subseteq \cdots$.   

The precise structure of $\K_n$, which
depends on $\ell$, was determined by Jones \cite{Jones2}.   It follows from
his results that $\K_n$ is a direct sum of 
matrix algebras $A_0, \dots, A_m$, where
$m \le \lfloor n/2 \rfloor$.
Moreover, we have $J_i = A_0 \oplus \cdots \oplus A_i$ for $1 \le i
\le m$.     Using 
the skein relations for the Kauffman bracket, together with
(\ref{braid}), one can check that for $i < m$, $s_n J_i
\not\subseteq J_i$ but $s_n J_i \subseteq J_{i+1}$.   

Let $\cal A$ denote the group algebra of $FB_{2n}(S^2)$, and
suppose $\phi \in \K_n$.
Recall that $\K_n$ is a quotient of the braid group algebra.  It
follows that $\K_n$ is also spanned by products of elements $r_i,
r_i^{-1} \in \Hom(n,n)$.     
By equation (\ref{braid}) it follows that
for any $\psi \in \K_n$ there exists $x \in \cal A$ such that
$ \psi\phi = x\phi $, 
and also $x \in \cal A$ such that 
$      \phi\psi = x\phi$.
Let $p_i \in \K_n$ denote the identity of $A_i$.
It follows that there exists an element $\pi_i
\in \cal A$ such that 
\[    \pi_i \phi = p_i\phi p_i .\]
Note that $\pi_i \maps \K_n \to A_i$ is the orthogonal projection down
to the summand $A_i$.

To prove the theorem it suffices to show that $\K_n$ is an irreducible
representation of $\cal A$.  Suppose that $M \subseteq \K_n$ is an
invariant subspace containing a nonzero vector $\phi$.  We will show
that $M = \K_n$.  

First, we can choose the least $i$ such that $\pi_i \phi \ne 0$, so
there is a nonzero vector in $A_i$ contained in $M$.   Since any
left or right multiplication in $A_i$ can be achieved by applying an
element of $\cal A$, it follows that all of $A_i$ lies in $M$.   By the
remarks above, as long as $i < m$, $\pi_{i+1} s_n \pi_i \in \cal A$ 
acts as a nonzero map from $A_i$ to $A_{i+1}$.    Thus there is a
nonzero vector in $A_{i+1}$ contained in $M$.  By induction we have
$A_i \oplus \cdots \oplus A_m \subseteq M$, and since $i$ was chosen as the
least with the given property it follows that
\[        A_i \oplus \cdots \oplus A_m = M  .\]
If $i = 0$ we are done.  Otherwise, since the framed braid group
representation is unitary, the orthogonal complement
\[        (A_i \oplus \cdots \oplus A_m)^\perp = A_0 \oplus \cdots
\oplus A_{i-1}  \]
must also be an invariant subspace.  Applying the above argument to
this subspace we conclude that it is all of $\K_n$, contradicting $M
\ne 0$.  \qed


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