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## 3.1 The definition

Definition 3.1   An n-singular integral homology sphere is a pair (M,L) where M is an integral homology sphere and is a unit-framed algebraically split ordered n-component link in M. Namely, the components Li of M are numbered 1 to n(ordered''), framed with framing (unit framed''), and the pairwise linking numbers between the different components of L are 0 (algebraically split''). We think of L as marking n sites for performing small modifications of M, each modification being the surgery on one of the components of L. Let us temporarily define to be the -module of all formal -linear combinations of n-singular integral homology spheres. A correction to the definition of will be given in Definition 3.2 below. Notice that , which we often simply denote by , is simply the space of all -linear combinations of integral homology spheres.

If is a framed link (presented as a union of two sublinks L1 and L2) in some 3-manifold M, we denote by (M,L1)L2the result of surgery2 of (M,L1) along L2. Namely, (M,L1)L2 is a pair (M',L1'), in which M' is the result of surgery of M along L2, and L1'is the image in M' of L1. Notice that if (M,L) is an (n+1)-singular integral homology sphere, then (M,L-Li)Li is again an n-singular integral homology sphere for any component Li of L.

We now wish to define the co-derivative map , whose adjoint will be the differentiation map for invariants:

Definition 3.2   Define on generators by , and extend it to be a -linear map . For later convinience, we want to set for any i, but the different i's may give different answers. We resolve this by redefining . Set

 (5)

We can now set (in the new )

The relations in equation (5) are called the co-differentiability relations''.

We can finally differentiate invariants using the adjoint . That is, if is a differentiable invariant of n-singular integral homology spheres (namely, which vanishes on the co-differentiability relations), let its derivative be . Iteratively, one can define multiple derivatives such us I(k) for any .

Definition 3.3   (Ohtsuki [Oh1] We say that an invariant I of integral homology spheres is of type nif its n+1st derivative vanishes. We say that it is of finite type if it is of type n for some natural number n.

Unravelling the definitions, we find that I is of type n precisely when for all integral homology spheres M and all unit-framed algebraically split (n+1)-component links L in M,

 (6)

where the sum runs on all sublinks L' of L (including the empty and full sublinks), |L'| is the number of components of L', and ML' is the result of surgery of M along L'. We will not use equation (6) in this paper.

#### Footnotes

... surgery2
We recall some basic facts about surgery in Section 3.2.1.

Next: 3.2 Preliminaries Up: 3. The case of Previous: 3. The case of
Dror Bar-Natan
2000-03-19