As we have already indicated in the introduction, the finite type theory
for knots (Vassiliev theory) is built around the notions of *n*-singular
knots, and differences between overcrossings and undercrossings. Let us
make those notions precise:

Notice that is simply the free -module generated by all (framed) knots.

Note that thanks to the co-differentiability relation, is well defined. It is called ``the co-derivative''. We denote the adjoint of by and call it ``the derivative''. It is a map .

The name ``derivative'' is justified by the fact that
for some
and
is by definition
the difference of the values of *V* on two ``neighboring'' *n*-singular
knots, in harmony with the usual definition of derivative for functions
on
.

When thinking about finite type invariants, it is convenient to have in mind the following ladders of spaces and their duals, printed here with the names of some specific elements that we will use later:

One may take the definition of a general ``theory of finite type
invariants'' to be the data in (2), with arbitrary
``*n*-singular objects'' replacing the *n*-singular knots. Much
of what we will say below depends only on the existance of the
ladders (2), or on the existance of certain natural
extensions thereof, and is therefore quite general.