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## 1.1 Singular knots, the co-differential , 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 , 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.

Definition 1.2   Let be the -module freely generated by all n-singular knots, modulo the following co-differentiability relation'':

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

Definition 1.3   Let be defined by resolving'' any one of the singular points in an (n+1)-singular knot in :

 (1)

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 .

Definition 1.4   An invariant of knots V (equivalently, a -linear functional on ) is said to be of finite type n if its (n+1)-st derivative vanishes, that is, if . (This definition is the analog of one of the standard definitions of polynomials 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:

 (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.

Next: 1.2 Constancy conditions, , Up: 1. The case of Previous: 1. The case of
Dror Bar-Natan
2000-03-19