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Definition 3.1
An
nsingular integral homology sphere is a pair
(
M,
L) where
M is an integral homology sphere and
is a unitframed algebraically split ordered
ncomponent link
in
M. Namely, the components
L_{i} 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
nsingular 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
L^{1} and L^{2}) in some 3manifold M, we denote by
(M,L^{1})_{L2}the result of surgery^{2}
of (M,L^{1}) along L^{2}. Namely,
(M,L^{1})_{L2} is a pair
(M',L^{1}'),
in which M' is the result of surgery of M along L^{2}, and L^{1}'is the image in M' of L^{1}. Notice that if (M,L) is an
(n+1)singular integral homology sphere, then
(M,LL_{i})_{Li} is again
an nsingular integral homology sphere for any component L_{i} of L.
We now wish to define the coderivative 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
codifferentiability relations''.
We can finally differentiate invariants using the adjoint
.
That is,
if
is a differentiable invariant of nsingular
integral homology spheres (namely, which vanishes on the
codifferentiability 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 unitframed 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 M_{L'} is
the result of surgery of M along L'. We will not use
equation (6) in this paper.
Footnotes
 ... surgery^{2}

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 BarNatan
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