Finite Type Invariants by the Species
Dror BarNatan
Date:
This Edition: Mar. 19, 2000;
First Edition: This is merely an unifinished draft;
Revision History: Section 6.
Institute of Mathematics
The Hebrew University
Giv'atRam, Jerusalem 91904
Israel
drorbn@math.huji.ac.il
Abstract:
We study the general theory of finite type invariants through the study
of examples.
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1. Introduction
In the late 1980s, Vassiliev [Va1,Va2]
suggested to study knot invariants by studying the space of all
knots. The first new ingredient thus added is the notion of
``neighboring'' knots, knots that differ in only one crossing, and
the idea that one should study how knot invariants change in the
``neighborhood'' of a given knot. This idea led to a definition
of a certain class of knot invariants, now known as ``Vassiliev''
or ``finite type'' invariants. By now more than 350 papers have been
written on the subject; see [BN2]. Vassiliev's original
definition (independently discovered by Goussarov [Go1,Go2] at roughly the same time) was formalized
in a different way by Birman and Lin [BL],
and later reinterpreted by analogy with multivariable calculus by
BarNatan [BN1].
The basic idea in BarNatan [BN1] was that
differences are ``cousins'' of derivatives, and hence one should
think of the difference between the values of a knot invariant
V on two neighboring knots as a ``derivative'' of the original
invariant. Repeating this, we find that iterated differences (as
considered by [Va1,Va2,Go1,Go2,BL,BN1])
of values of V in the neighborhood of some knot should be thought of
as multiple derivatives. A ``Vassiliev'' or ``finite type'' invariant
of type m is then the analog of a polynomial  an invariant whose
m+1st derivatives, or m+1st iterated differences, vanishes.
Clearly, these ideas are very general, and knots (and even topology in
general) are just a particular case. Whenever an appropriate notion of
``neighborhood of an object'' exists, one can talk about finite type
invariants of such objects. This leads to many different ``species''
of finite type invariants. Let us mention just a few:
 The usual notion of nearness of knots, knots that differ at only
a single crossing, leads to the usual Vassiliev invariants.
 Similarly, one can define ``Vassiliev'' invariants of braids, links,
tangles, knotted graphs, etc.
 Goussarov [Go3] has also an alternative
notion of a neighborhood of a knot (or link), defined by
``interdependent modifications''. This notion leads to a different
(though at the end, equivalent) theory of finite type invariants of
knots and links.
 Two algebraically split links (links whose linking numbers all vanish)
can be considered neighboring if they differ by the simultaneous flip of
two opposite crossings between two given components (such a double
flip preserves linking numbers, whereas a single flip doesn't). See
Figure 1. This leads to a little known but probably
interesting theory of finite type invariants of algebraically split
links.
Figure 1:
A move that preserves linking numbers.

 Several authors [] have considered several notions of finite type
invariants of plane curves.
 Ohtsuki [Oh1] considers two integral
homology spheres to be neighboring (roughly) if they differ by a single
surgery. This leads to a notion of finite type invariants of integral
homology spheres. Several variants of his definition were considered
in [Ga1,GL1,GO1,GL2,GO2,Ga2,GL3]
A finite type theory automatically comes bundled with several spaces
that play a significant role in it. In the best known case of knots,
these are the spaces of chord diagrams, 4T relations, weight systems,
chord diagrams modulo 4T relations, relations between 4T relations,
and a few lesser known spaces that should probably be better known. There
is a ``general theory of finite type invariants'', defined in terms of
these spaces, in which one attempts to classify finite type invariants by
first classifying their potential mth derivatives, and then by studying
which of those potential derivatives can actually be integrated to an
honest invariant. I should say that though this ``general theory'' is
rather small, it is also rather interesting (with the most interesting
parts developed by M. Hutchings [Hu] and
private communication), and insufficiently well known even in the
case of the usual finite type invariants of knots.
The purpose of this paper is twofold:
 1.
 To state (and propagate) this general theory of finite type
invariants of anything. Namely, to construct, name, and study
the relationships between those spaces that come automatically
with every finite type theory, especially from the perspective of
the integration theory of ``weight systems''. We first do it in
Section 1 on a well known example, the original finite
type theory of knots. We then extract some general features from this
example and give them general names; this is done in the rather short
Section 2.
 2.
 To list many of the currently known finite type theories, and
figure out (to the degree that is now possible) what these associated
spaces are on a species by species basis. Our list takes the form of a
``classification''^{1}.
The top subdivision is into the classes of ``Knotted Objects'',
``3Manifolds'' and ``Plane Curves'', and these classes are
described in Sections 4, 5
and 6, respectively.
Acknowledgement: I wish to thank Mike Hutchings for the inspiration
to write this paper and for telling me his ideas about integration
theory. I also wish to thank E. Appleboim and H. Scolnicov for many
ideas and conversations.
1. The case of knots
As we have already indicated in the introduction, the finite type theory
for knots (Vassiliev theory) is built around the notions of nsingular
knots, and differences between overcrossings and undercrossings. Let us
make those notions precise:
Definition 1.1
An
nsingular 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.
Figure 2:
A singular point.

Definition 1.2
Let
be the
module freely generated by all
nsingular
knots, modulo the following ``codifferentiability 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 codifferentiability relation,
is well
defined. It is called ``the coderivative''. 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'' nsingular
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
``nsingular objects'' replacing the nsingular 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.
As promised in the introduction, we study invariants of type n by
studying their nth derivatives. Clearly, if V is of type n and
,
then
(``W is a constant''). Glancing
at (2), we see that W descends to a linear functional,
also called W, on
:
Definition 1.5
We call
the space of ``
nsymbols'' associated with the ladders
in (
2). (The name is inspired by the theory of differential
operators, where the ``symbol'' of an operator is essentially its
equivalence class modulo lower order operators. The symbol is responsible
for many of the properties of the original operator, and for many
purposes, two operators that have the same symbol are ``the same''.)
We denote the projection mapping
that maps every
singular knot to its symbol by
.
The following classical proposition (see e.g. [BN1,Bi,BL,Go1,Go2,Ko1,Va1,Va2] identifies the space of nsymbols in our case:
Proposition 1.6
The space
of
nsymbols for (
2) is
canonically isomorphic to the space
of
nchord diagrams,
defined below.
Definition 1.7
An
nchord diagram is a choice of
n pairs of
distinct points on an oriented circle, considered up to orientation
preserving homeomorphisms of the circle. Usually an
nchord diagram is
simply drawn as a circle with
n chords (whose ends are the
n pairs),
as in the 5chord example in Figure
3. The space
is the space of all formal
linear combinations of
nchord diagrams.
Figure 3:
A chord diagram.

Next, we wish to find conditions that a ``potential top derivative'' has
to satisfy in order to actually be a top derivative. More precisely, we
wish to find conditions that a functional
has
to satisfy in order to be
for some invariant V. A first
condition is that W must be ``integrable once''; namely, there has to
be some
with
.
Another quick
glance at (2), and we see that W is integrable once iff
it vanishes on
,
which is the same as requiring that Wdescends to
(there should be no confusion
regarding the identities of the 's involved). Often elements of
are refered to as ``weight systems''. A more accurate
name would be ``onceintegrable weight systems''.
We see that it is necessary to understand
.
In
Figure 4 we show a family of members of
,
the
``Topological 4Term'' (T4T) relations. Figure 5 explains
how they arise from ``lassoing a singular point''. The following theorem
says that this is all:
Figure 4:
A Topological 4Term (T4T) relation. Each of the four graphics in the
picture represents a part of an nsingular knot (so there are n2 additional singular points not shown), and, as usual in knot theory, the
4 singular knots in the equation are the same outside the region shown.

Figure 5:
Lassoing a singular point: Each of the graphics represents an
(n1)singular knot, but only one of the singularities is explicitly
displayed. Start from the leftmost graphic, pull the ``lasso'' under
the displayed singular point, ``lasso'' the singular point by crossing
each of the four arcs emenating from it one at a time, and pull the
lasso back out, returning to the initial position. Each time an arc is
crossed, the difference between ``before'' and ``after'' is the
coderivative of an nsingular knot (up to signs). The four
nsingular knot thus obtained are the ones making the Topological
4Term relation, and the coderivative of their signed sum is the
difference between the first and the last (n1)singular knot shown
in this figure; namely, it is 0.

Pushing the T4T relations down to the level of symbols, we get the
wellknown 4T relations, which span
:
(see
e.g. [BN1])
We thus find that
,
as usual in the theory of finite type invariants of knots.
We have so far found that if V is a typen invariant, then
is a linear functional on
.
A question arises
whether every linear functional on
arises in this way. At least
if the ground ring is extended to
,
the answer is positive:
Theorem 2 (The Fundamental Theorem of
Finite Type Invariants, Kontsevich [
Ko1])
Over
,
for every
there exists a type
ninvariant
V with
.
In other words, every onceintegrable
weight system is fully integrable.
The problem with the Fundamental Theorem is that all the proofs we have
for it are somehow ``transcendental'', using notions from realms outside
the present one, and none of the known proofs settles the question over
the integers (see [BS]). In this section
we describe what appears to be the most natural and oldest approach to
the proof, having been mentioned already in [Va1,BL]. Presently, we are stuck and the socalled
``topological'' approach does not lead to a proof. But it seems to me that
it's worth studying further; when something natural fails, there ought to
be a natural reason for that, and it would be nice to know what it is.
The idea of the topological approach is simple: To get from W to
V, we need to ``integrate'' n times. Let's do this one integral at
a time. By the definition of
,
we know that we can integrate
once and find
so that
.
Can
we work a bit harder, and find a ``good'' W^{1}, so that there would
be a
with
? Proceeding like
that and assuming that all goes well along the way, we would end with a
with
,
as required. Thus we are
naturally lead to the following conjecture, which implies the Fundamental
Theorem by the backwardinductive argument just sketched:
Conjecture 1
Every onceintegrable invariant of
nsingular knots also twice
integrable. Glancing at (
2), we see that this is the same
as saying that
.
This conjecture is somewhat stronger than Theorem 2.
Indeed, Theorem 2 is equivalent to
Conjecture 1 restricted to the case when the given
invariant has some (possibly high) derivative identically equal to 0(exercise!). But it is hard to imagine a topological proof of the
restricted form of Conjecture 1 that would not prove
it in full.
The difficulty in Conjecture 1 is that it's hard
to say much about
.
In [Hu],
Michael Hutchings was able to translate the statement
to an easierlooking
combinatorialtopological statement, which is implied by
and perhaps equivalent to an even simpler fully combinatorial
statement. Furthermore, Hutchings proved the fully combinatorial statement
in the analogous case of finite type braid invariants, thus proving
Conjecture 1 and Theorem 2
(over
)
in that case, and thus proving the viability of his
technique.
Hutchings' first step was to write a chain of isomorphisms reducing
to something more manageable. Our next step
will be to introduce all the spaces participating in Hutchings' chain.
First, let us consider the space of all T4T relations:
Definition 1.8
Let
be the
module generated by all (framed) knots
having
n2 singularities as in Definition
1.1,
and plus one additional ``Topological Relator'' singularity
that locally looks like the image in Figure
6,
modulo the same codifferentiability relations as in
Definition
1.2. Define
in the same way as for knots, using equation (
1). Finally,
define
by mapping the topological relator to
the topological 4term relation, the 4term alternating sum inside the
paranthesis in Figure
4.
Figure 6:
The ``Topological Relator'' singularity.

The spaces
form a ladder similar to the one
in (2), and, in fact, they combine with the ladder
in (2) to a single commutative diagram:

(3) 
In this language, Stanford's theorem (Theorem 1) says
that all L shapes in the above diagram (compositions
of
``down'' followed by ``right'') are exact.
Just like singular knots had symbols which were simplar combinatorial
objects (chord diagrams), so do toplogical relators have combinatorial
symbols:
Definition 1.9
Let
,
and let
be the projection map.
The following proposition is proved along the same lines as the standard
proof of Proposition 1.6.
Proposition 1.10
is canonically isomorphic to the space
spanned by all ``relator symbols'', chord diagrams with
n2 chords and
one
piece corresponding to the special
singularity of Definition
1.8. An example appears in
Figure
7.
Figure 7:
A ``relator symbol''.

We need to display one additional commutative diagram before we can come to
Hutchings' chain of isomorphisms:

(4) 
In this diagram,
is the ``symbol level'' version of b, and
is induced by
in the usual manner. It
can be described combinatorially by
Hutchings' chain of isomorphisms is the following chain of equalities and
maps: (here the symbol
means that the space below is a subspace of
the space above, and the symbol
means that the space below is a
subquotient of the space above)
Theorem 3 (Hutchings [
Hu])
All maps in the above chain are isomorphisms. In particular,
.
Proof. Immediate from diagrams (3)
and (4).
It doesn't look like we've achieved much, but in fact we did, as it
seems that
is easier to digest than the
original space of interest,
.
The point
is that
lives fully in the combinatorial realm, being
essentially the space of all relations between 4T relations at the
symbol level. Similarly,
is the space of projections to the
symbol level of relation between 4T relations, and hence we have shown
Corollary 1.11
Conjecture
1 is equivalent to the
statement ``every relation between 4
T relations at the symbol level has a
lift to the topological level''.
An obvious approach to proving Conjecture 1 thus
emerges:
 Combinatorial step: Find all relations between 4T relations at
the symbols level; that is, find a generating set for
.
 For every relation found in the combinatorial step, show that it
lifts to the topological level.
So far, the problem with this approach appears to be in the combinatorial
step. There is a conjectural generating set
for
.
Every element in
indeed
has a lifting to ,
but we still don't know if
indeed generates
.
We state these facts very briefly; more
information can be found in [Hu] and
in [BS].
Definition 1.12
Define
by
As usual, each graphic in the above formula represents a large number of
elements of
,
obtained from the graphic by the addition
of
n3 chords (first graphic), or
n5 chords (second graphic), or
n4 chords (third graphic), or
n2 chords (fourth graphic). Define
also
by
A parallel of Conjecture 2 for braids was proven by Hutchings
in [Hu].
Exercise 1.13
Find a space
and maps
and
that
fit into a commutative diagram,
and hence show that the relations in
all lift to
.
Question 1.14
Is the sequence
related to Kontsevich's graph
cohomology [
Ko2]?
In summary, we have introduced and studied the following objects, spaces,
and maps:
Objects:
is the space of all framed oriented knots in
an oriented
.
More precisely, it is the free
module
generated by framed oriented knots in an oriented
.
The nCubes:
is the free
module generated
by framed oriented knots in an oriented
,
that have
precisely n double point singularities
as in
Figure 2, modulo the codifferentiability relation
of Definition 1.2,
The CoDerivative: The coderivative
is the map
defined by
The Cube Ladder and Finite
Type Invariants:
The nSymbols: The space of nsymbols
is the space
of nchord diagrams, as in
Figure 3.
The Relator Ladder: The relator ladder is the ladder
(see Equation 3), of singular knots with
exactly one ``Topological Relator'' singularity as in
Figure 6.
The Primary
Integrability Constraints: The primary integrability constraints are the images of
the relators via the map b; that is, they are the Topological 4Term
relations of Figure 4.
The Relator Symbols and
the SymbolLevel Relations: The relator symbols are diagrams of the kind appearing in
Figure 7.
The OnceReduced Symbol
Space and Once Integrable Weight Systems:
The Inductive
Problem:
The Lifting Problem:
Generic SymbolLevel
Redundencies:
The ObjectLevel
Redundencies:
The Redundency
Problem:
2. The General Theory of Finite Type Invariants
TBW.
TBW.
3. The case of integral homology spheres
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.
3.2 Preliminaries
3.2.1 Surgery and the Kirby calculus
Figure 8:
The Left Twist (LT).

TBW
3.2.3 The triple linking numbers
TBW
3.3 Constancy conditions or
Definition 3.4
Let
be the unital commutative
algebra over
generated by symbols
Y_{ijk} for distinct
indices
,
modulo the anticyclicity relations
Y_{ijk}=
Y^{1}_{jik}=
Y_{jki}.
Warning 3.5
Below we will mostly regard
as an
module, and not as an algebra. Thus we will only use the
product of
as a convenient way of writing certain elements
and linear combinations of elements. The subspaces of
that we
will consider will be subspaces in the linear sense, but not ideals or
subalgebras, and similarly for quotients and maps from or to
.
It is easy to define a map
.
For an nlink L set
It follows from Section 3.2.3 that this definition descends to
the quotient of
by the coderivatives of (n+1)links.
Theorem 4
The thus defined map
is an isomorphism.
Figure 9:
A 3mask.

Figure 10:
The coderivative of a 3mask.

Figure 11:
The Bundle Left Twist (BLT) is the same as the Left Twist, only
that the strands within each ``bundle'' are not twisted internally.

TBW.
3.4 Integrability conditions or
Figure 12:
Undoing a Bundle Left Twist one crossing at a time.

Figure 13:
The Total Twist Relation (TTR).

Figure 14:
The Total Twist Relation (TTR).

Figure 15:
The Monster

Figure 16:
Lassoing a Borromean link.

(The last equality holds because in the two error terms,
Y_{rab}Y_{rgb} and Y_{rgb}, the component p is unknotted).
Now reduce the component r using the total twist relation. Only the first
term is affected, and 3 of the 6 terms that are produced from its
reduction cancel against the 3 remaining terms of the above equation. The
result is:
The last term here drops out because in it the component r is unknotted,
and so the end result is
.
In graphical terms,
this is precisely the graph I! Cyclically permuting the roles of r,
g, and b, we find that we have proven the IHX relation.
An image from http://www.best.com/~abacus/oro/ouroboros.html.
4. The class of Knotted Objects
TBW
TBW
TBW
5. The class of 3Manifolds
6. The class of Plane Curves

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 D. BarNatan,
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 ,
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,
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 ,
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 ,
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 M. Hutchings,
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 ,
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 T. Ohtsuki,
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 St1
 T. Stanford,
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 Va1
 V. A. Vassiliev,
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Finite Type Invariants by the Species
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Footnotes
 ... ``classification''^{1}

classification (klàs´efk´shen), in biology,
the systematic categorization of organisms. One aim of modern
classification, or systematics, is to show the evolutionary
relationships among organisms. The broadest division of organisms
is into kingdoms, traditionally twoAnimalia (animals) and Plantae
(plants). Widely accepted today are three additional kingdoms:
the Protista, comprising protozoans and some unicellular algae;
the Monera, bacteria and bluegreen algae; and the Fungi. From
most to least inclusive, kingdoms are divided into the following
categories: phylum (usually called division in botany), class, order,
family, genus, and species. The species, the fundamental unit of
classification, consists of populations of genetically similar,
interbreeding or potentially interbreeding individuals that share
the same gene pool (collection of inherited characteristics whose
combination is unique to the species). Copyright ©1995
by Columbia University Press. All rights reserved.
 ... surgery^{2}

We recall some basic facts about surgery in
Section 3.2.1.
Dror BarNatan
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