We have so far found that if *V* is a type-*n* invariant, then
is a linear functional on
.
A question arises
whether every linear functional on
arises in this way. At least
if the ground ring is extended to
,
the answer is positive:

The problem with the Fundamental Theorem is that all the proofs we have for it are somehow ``transcendental'', using notions from realms outside the present one, and none of the known proofs settles the question over the integers (see [BS]). In this section we describe what appears to be the most natural and oldest approach to the proof, having been mentioned already in [Va1,BL]. Presently, we are stuck and the so-called ``topological'' approach does not lead to a proof. But it seems to me that it's worth studying further; when something natural fails, there ought to be a natural reason for that, and it would be nice to know what it is.

The idea of the topological approach is simple: To get from *W* to
*V*, we need to ``integrate'' *n* times. Let's do this one integral at
a time. By the definition of
,
we know that we can integrate
once and find
so that
.
Can
we work a bit harder, and find a ``good'' *W*^{1}, so that there would
be a
with
? Proceeding like
that and assuming that all goes well along the way, we would end with a
with
,
as required. Thus we are
naturally lead to the following conjecture, which implies the Fundamental
Theorem by the backward-inductive argument just sketched:

This conjecture is somewhat stronger than Theorem 2. Indeed, Theorem 2 is equivalent to Conjecture 1 restricted to the case when the given invariant has some (possibly high) derivative identically equal to 0(exercise!). But it is hard to imagine a topological proof of the restricted form of Conjecture 1 that would not prove it in full.

The difficulty in Conjecture 1 is that it's hard to say much about . In [Hu], Michael Hutchings was able to translate the statement to an easier-looking combinatorial-topological statement, which is implied by and perhaps equivalent to an even simpler fully combinatorial statement. Furthermore, Hutchings proved the fully combinatorial statement in the analogous case of finite type braid invariants, thus proving Conjecture 1 and Theorem 2 (over ) in that case, and thus proving the viability of his technique.

Hutchings' first step was to write a chain of isomorphisms reducing
to something more manageable. Our next step
will be to introduce all the spaces participating in Hutchings' chain.
First, let us consider the space of all *T*4*T* relations:

The spaces form a ladder similar to the one in (2), and, in fact, they combine with the ladder in (2) to a single commutative diagram:

In this language, Stanford's theorem (Theorem 1) says that all L shapes in the above diagram (compositions of ``down'' followed by ``right'') are exact.

Just like singular knots had symbols which were simplar combinatorial objects (chord diagrams), so do toplogical relators have combinatorial symbols:

The following proposition is proved along the same lines as the standard proof of Proposition 1.6.

We need to display one additional commutative diagram before we can come to Hutchings' chain of isomorphisms:

In this diagram,
is the ``symbol level'' version of *b*, and
is induced by
in the usual manner. It
can be described combinatorially by

Hutchings' chain of isomorphisms is the following chain of equalities and maps: (here the symbol means that the space below is a subspace of the space above, and the symbol means that the space below is a sub-quotient of the space above)

*Proof. * Immediate from diagrams (3)
and (4).

It doesn't look like we've achieved much, but in fact we did, as it
seems that
is easier to digest than the
original space of interest,
.
The point
is that
lives fully in the combinatorial realm, being
essentially the space of all relations between 4*T* relations at the
symbol level. Similarly,
is the space of projections to the
symbol level of relation between 4*T* relations, and hence we have shown

An obvious approach to proving Conjecture 1 thus emerges:

- Combinatorial step: Find all relations between 4
*T*relations at the symbols level; that is, find a generating set for . - For every relation found in the combinatorial step, show that it
lifts to the topological level.

So far, the problem with this approach appears to be in the combinatorial step. There is a conjectural generating set for . Every element in indeed has a lifting to , but we still don't know if indeed generates . We state these facts very briefly; more information can be found in [Hu] and in [BS].

As usual, each graphic in the above formula represents a large number of elements of , obtained from the graphic by the addition of

A parallel of Conjecture 2 for braids was proven by Hutchings in [Hu].

and hence show that the relations in all lift to .