Kontsevich's integral is a far-reaching generalization of the Gauss integral for the linking number, and provides a tool to construct the universal Vassiliev invariant of a knot. In fact, any Vassiliev knot invariant can be derived from it.
To construct the Kontsevich integral, represent the three-dimensional space 
as a direct
product of a complex line 
with coordinate 
and a real line 
with coordinate 
. The integral is defined for Morse
knots, i.e., knots 
embedded in 
in such a way that the coordinate 
is a Morse function on 
, and its values belong to the graded
completion 
of the algebra of chord diagrams 
.
The Kontsevich integral 
of the knot 
is defined as
 |
(1)
|
where the ingredients of this formula have the following meanings. The real numbers 
and 
are the minimum and the maximum of the function 
on 
.
The integration domain is the 
-dimensional simplex 
divided by the critical
values into a certain number of connected components. For example, for the embedding
of the unknot and 
(left figure), the corresponding integration domain has six connected components,
illustrated in the right figure above.
The number of addends in the integrand is constant in each connected component of the integration domain, but can be different for different
components. In each plane 
, choose an unordered pair of distinct points
and 
on 
so that 
and 
are continuous functions. Denote by 
the set of such pairs for 
, ..., 
, then the integrand is the sum over all choices of 
. In the example above, for the component 
, we have
only one possible pair of points on the levels 
and 
. Therefore, the sum over 
for this component consists of only one addend.
In contrast, in the component 
, we still
have only one possibility for the level 
, but the plane 
intersects our knot
in four points. So we have 
possible pairs 
, and the total number of addends
is six (see the picture below).
For a pairing 
the symbol '
'
denotes the number of points 
or 
in 
where the coordinate 
decreases along the orientation
of 
.
Fix a pairing 
,
consider the knot 
as an oriented circle, and connect the points 
and 
by a chord to obtain a chord diagram with 
chords. The corresponding element of the algebra 
is denoted 
. In the picture above, one of the possible pairings, the
corresponding chord diagram with the sign 
, and the number of addends
of the integrand (some of which are equal to zero in 
due to a one-term relation)
are shown for each connected component.
Over each connected component, 
and 
are smooth functions
in 
.
By 
we mean the pullback of this form to the integration
domain of variables 
, ..., 
. The integration domain is considered with the manifold
orientation of the space 
defined by the natural order of the coordinates 
, ..., 
.
By convention, the term in the Kontsevich integral corresponding to 
is the (only) chord diagram
of order 0 with coefficient one. It represents the unit of the algebra 
.
The Kontsevich integral is convergent thanks to one-term relations. It is invariant under deformations
of the knot in the class of Morse knots. Unfortunately,
the Kontsevich integral is not invariant under deformations that change the number
of critical points of the function 
. However, the formula shows how the integral changes under
such deformations:
In the above equation, the graphical arguments of 
represent two embeddings of an arbitrary knot, differing only
in the illustrated fragment,
is the hump (i.e., the unknot embedded in 
in the specified way; illustrated
above), and the product is the product in the completed algebra 
of chord diagrams.
The last equality allows the definition of the universal
Vassiliev invariant by the formula
 |
(2)
|
where 
denotes the number of critical points of 
and quotient means division in the algebra 
according to the rule 
. The universal
Vassiliev invariant 
is invariant under an arbitrary deformation
of 
.
Consider a function 
on the set of chord diagrams
with 
chords satisfying one- and four-term relations
(a weight system). Applying this function to the
universal Vassiliev invariant 
, we get a numerical knot invariant.
This invariant will be a Vassiliev invariant
of order 
,
and any Vassiliev invariant can be obtained
in this way.
The Kontsevich integral behaves in a nice way with respect to the natural operations on knots, such as mirror reflection, changing the orientation of the knot, and mutation of knots. In a proper normalization it is multiplicative under the connected sum of knots:
 |
(3)
|
where 
.
For any knot 
the coefficients in the expansion of 
over an arbitrary basis consisting of chord
diagrams are rational (Kontsevich 1993, Le and Murakami 1996).
The task of computing the Kontsevich integral is very difficult. The explicit expression of the universal Vassiliev invariant 
is currently known only for the unknot,
 |  |  |
(4)
|
 |  |  |
(5)
|
(Bar-Natan et al. 1995). Here, 
are modified
Bernoulli numbers, i.e., the coefficients of the Taylor
series
 |
(6)
|
(
,
,
...; OEIS A057868), and 
are the wheels, i.e., diagrams of the form
The linear combination is understood as an element of the algebra of Chinese characters 
, which is isomorphic to the algebra
of chord diagrams 
. Expressed through chord diagrams,
the beginning of this series looks as follows:
The Kontsevich integral was invented by Kontsevich (1993), and detailed expositions can be found in Arnol'd (1994), Bar-Natan (1995), and Chmutov and Duzhin (2000).
