1.1 Singular knots, the co-differential $\delta$, and finite type invariants

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:

Definition 1.1   An n-singular knot is an oriented knot in an oriented ${\mathbb R}^3$, which is allowed to have n singular points that locally look like the image in Figure 2. For simplicity in the later parts of this section, we only consider framed (singular or not) knots, and always use blackboard framing when a knot projection or a part of a knot projection is drawn.

  

Figure 2: A singular point.

Definition 1.2   Let ${\mathcal K}_n$ be the ${\mathbb Z}$-module freely generated by all n-singular knots, modulo the following ``co-differentiability relation'':

$$\if ny \smash{\makebox[0pt]{\hspace{-0.5in} \raisebox{8pt}{... ...-8pt}{ \input draws/codiff.tex } \hspace{-1.9mm} \end{array}$$

Notice that ${\mathcal K}_0={\mathcal K}$ is simply the free ${\mathbb Z}$-module generated by all (framed) knots.

Definition 1.3   Let $\delta:{\mathcal K}_{n+1}\to{\mathcal K}_n$ be defined by ``resolving'' any one of the singular points in an (n+1)-singular knot in ${\mathcal K}_{n+1}$:

$$\if ny \smash{\makebox[0pt]{\hspace{-0.5in} \raisebox{8... ...pt}{ \input draws/deltadef.tex } \hspace{-1.9mm} \end{array}$$ (1)

Note that thanks to the co-differentiability relation, $\delta$ is well defined. It is called ``the co-derivative''. We denote the adjoint of $\delta$ by $\partial$ and call it ``the derivative''. It is a map $\partial:{\mathcal K}_n^\star\to{\mathcal K}_{n+1}^\star$.

The name ``derivative'' is justified by the fact that $(\partial V)(K)$for some $V\in{\mathcal K}_n^\star$ and $K\in{\mathcal K}_{n+1}$ 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 ${\mathbb R}^d$.

Definition 1.4   An invariant of knots V (equivalently, a ${\mathbb Z}$-linear functional on ${\mathcal K}$) is said to be of finite type n if its (n+1)-st derivative vanishes, that is, if $\partial^{n+1}V\equiv 0$. (This definition is the analog of one of the standard definitions of polynomials on ${\mathbb R}^d$).

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:

$$\begin{array}{cccccccccr} \ldots\longrightarrow{\hspace{... ...^{n+1}V\equiv 0 & & \partial^nV=W & & & & & & V\ \end{array}$$ (2)

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.