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An integral giving the universal Vassiliev knot invariant. Any Vassiliev knot invariant [[#References|[a6]]] can be derived from it. The integral is defined for a knot $K$ (cf. also [[Knot theory|Knot theory]]) embedded in the three-dimensional space $ \mathbf{R} ^ { 3 } =  \mathbf{C} _ { z } \times  \mathbf{R} _ { t }$ in such a way that the coordinate $t$ is a [[Morse function|Morse function]] on $K$ (all critical points are non-degenerate and all critical levels are different). Its values belong to the graded completion $\overline{\mathcal{A}}$ of the algebra of chord diagrams $\mathcal{A}$ defined below.

The Kontsevich integral $Z ( K )$ is an iterated integral given by the formula:

\begin{equation*} \sum _ { m = 0 } ^ { \infty } \frac { 1 } { ( 2 \pi i ) ^ { m } } \int _ { T } \sum _ { P = \{ ( z _ { j } , z _ { j } ^ { \prime } ) \} } ( - 1 ) ^ { \downarrow } D _ { P } \bigwedge _ { j = 1 } ^ { m } \frac { d z _ { j } - d z _ { j } ^ { \prime } } { z _ { j } - z _ { j } ^ { \prime } }. \end{equation*}

The ingredients of this formula are as follows:

1) The real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201009.png"/> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010010.png"/> are the minimum and the maximum of the function $t$ on $K$.

2) The integration domain $T$ is the $m$-dimensional simplex $t _ { \operatorname { min } } &lt; t _ { 1 } &lt; \ldots &lt; t _ { m } &lt; t _ { \operatorname { max } }$ divided by the critical values into a certain number of connected components; $T = \{ ( t _ { 1 } , \dots , t _ { m } ) : t _ { \operatorname { min } } &lt; t _ { 1 } &lt; \ldots &lt; t _ { m } &lt; t _ { \operatorname { max } } ,\; t_{j} \text{ non} \square \text{critical} \}$.

3) The number of summands in the integrand is constant in each connected component of the integration domain, but can be different for different components. Each plane $\{ t = t _ { j } \} \subset \mathbf{R} ^ { 3 }$ intersects the knot $K$ in some number, say $n_j$, of points. The numbers $n_j$ are constants if $( t _ { 1 } , \dots , t _ { m } )$ belongs to a fixed connected component of the integration domain, but in general they can be different for different components. Choose one of $\left( \begin{array} { c } { n_j } \\ { 2 } \end{array} \right)$ unordered pairs of distinct points $( z _ { j } , t _ { j } )$ and $( z _ { j } ^ { \prime } , t _ { j } )$ on $\{ t = t_j \} \cup K$ for each $j$. Denote by $P = \{ ( z _ { j } , z _ { j } ^ { \prime } ) \}$ a set of such pairs for all $j = 1 , \ldots , m$. The integrand is the sum over all choices $P$.

4) For a pairing $P$ the symbol $\downarrow$ denotes the number of points $( z _ { j } , t _ { j } )$ or $( z _ { j } ^ { \prime } , t _ { j } )$ in $P$ where the coordinate $t$ decreases along the orientation of $K$.

5) Fix a pairing $P$. Consider the knot $K$ as an oriented circle and connect the points $( z _ { j } , t _ { j } )$ and $( z _ { j } ^ { \prime } , t _ { j } )$ by a chord. One obtains a chord diagram with $m$ chords. The corresponding element of the algebra $\mathcal{A}$ is denoted by $D _ { P }$. The algebra of chord diagrams $\mathcal{A}$ is the [[Graded algebra|graded algebra]] $\mathcal{A} = \mathcal{A} _ { 0 } \oplus \mathcal{A} _ { 1 } \oplus \ldots$. The linear space $\mathcal{A} _ { m }$ is generated over $\mathbf{C}$ by all chord diagrams with $m$ chords considered modulo the relations of the following two types:

One-term relations:

<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/k120100a.gif" style="border:1px solid;"/>

Figure: k120100a

(here and below, the dotted arcs suggest that there might be further chords attached to their points, while on the solid portions of the circle all the endpoints are explicitly shown); and

Four-term relations:

<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/k120100b.gif" style="border:1px solid;"/>

Figure: k120100b

for an arbitrary fixed position of $( n - 2 )$ chords (which are not drawn here) and the two additional chords positioned as shown in the picture.

The multiplication in $\mathcal{A}$ is defined by the connected sum of chord diagrams, which is well-defined thanks to the four-term relations. In fact $\mathcal{A}$ is even a [[Hopf algebra|Hopf algebra]] but the co-multiplication is not needed here.

6) Over each connected component, $z_j$ and $z _ { j } ^ { \prime }$ are smooth functions in $t_j$. With some abuse of notation,

\begin{equation*} \bigwedge _ { j = 1 } ^ { m } \frac { d z _ { j } - d z _ { j } ^ { \prime } } { z _ { j } - z _ { j } ^ { \prime } } \end{equation*}

is to be interpreted as the pullback of this form to the integration domain of variables $t _ { 1 } , \ldots , t _ { m }$. The integration domain is considered with the positive orientation of the space $\mathbf{R} ^ { m }$ defined by the natural order of the coordinates $t _ { 1 } , \ldots , t _ { m }$.

7) By convention, the term in the Kontsevich integral corresponding to $m = 0$ is the (only) chord diagram of order $0$ (without chords) with coefficient one. It represents the unit of the algebra $\mathcal{A}$.

The Kontsevich integral is convergent thanks to the one-term relations. It is invariant under deformations of a knot $K$ 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 $t$. However, the following formula shows how the integral changes under such deformations:

<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/k120100c.gif" style="border:1px solid;"/>

Figure: k120100c

Here, the first and the third pictures depict an arbitrary knot, differing only in the fragment shown, while the second picture represents the unknot embedded in $\mathbf{R} ^ { 3 }$ in the specified way and the product is the product in the completed algebra $\overline{\mathcal{A}}$ of chord diagrams. The last equality allows one to define the universal Vassiliev invariant by the formula

<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/k120100d.gif" style="border:1px solid;"/>

Figure: k120100d

Here, $c$ denotes the number of critical points of $K$ and the quotient means division in the algebra $\overline{\mathcal{A}}$:

\begin{equation*} ( 1 + a ) ^ { - 1 } = 1 - a + a ^ { 2 } - a ^ { 3 } +\dots . \end{equation*}

The universal Vassiliev invariant $\tilde{Z} ( K )$ is invariant under an arbitrary deformation of $K$.

Consider a function $w$ on the set of chord diagrams with $m$ chords satisfying one- and four-term relations. Applying this function to the universal Vassiliev invariant $w ( \widetilde{Z} ( K ) )$, one obtains a numerical knot invariant. This invariant will be a Vassiliev invariant of order $m$ and any Vassiliev invariant can be obtained in this way. The Kontsevich integral is extremely complicated. For a long time even the Kontsevich integral of the unknot

<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/k120100e.gif" style="border:1px solid;"/>

Figure: k120100e

was unknown. The conjecture about it appeared only recently [[#References|[a7]]]. D. Bar-Natan, T. Le and D. Thurston proved the conjecture but the preprint is still in preparation (1999).

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. It is multiplicative under the connected sum of knots (because it is a group-like element in the Hopf algebra $\overline{\mathcal{A}}$). The claim that the coefficients of $Z ( K )$ are rational, [[#References|[a5]]], was proved in [[#References|[a4]]].

The Kontsevich integral was invented by M. Kontsevich [[#References|[a5]]]. See [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]] for detailed expositions of the relevant theory.

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top"> V.I. Arnol'd, "Vassiliev's theory of discriminants and knots" , ''First European Congress of Mathematicians (Paris)'' , Birkhäuser (1992) pp. 3–29 {{ZBL|0869.57006}}</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top"> D. Bar-Natan, "On the Vassiliev knot invariants" ''Topology'' , '''34''' (1995) pp. 423–472 {{MR|}} {{ZBL|0898.57001}} </td></tr>
<tr><td valign="top">[a3]</td> <td valign="top"> S.V. Chmutov, S.V. Duzhin, "The Kontsevich integral" ''Acta Applic. Math.'' (to appear) (available via anonymous ftp: pier.botik.ru, file: pub/local/zmr/ki.ps.gz) {{MR|1837618}} {{ZBL|0980.57006}} </td></tr>
<tr><td valign="top">[a4]</td> <td valign="top"> T.Q.T. Le, J. Murakami, "The universal Vassiliev-Kontsevich invariant for framed oriented links" ''Compositio Math.'' , '''102''' (1996) pp. 42–64 {{MR|1394520}} {{ZBL|0851.57007}} </td></tr>
<tr><td valign="top">[a5]</td> <td valign="top"> M. Kontsevich, "Vassiliev's knot invariants" ''Adv. Soviet Math.'' , '''16''' (1993) pp. 137–150 {{MR|1237836}} {{ZBL|}} </td></tr>
<tr><td valign="top">[a6]</td> <td valign="top"> V.A. Vassiliev, "Theory of singularities and its applications" V.I. Arnol'd) (ed.) , ''Advances in Soviet Math.'' , '''1''' , Amer. Math. Soc. (1990) pp. 23 –69</td></tr>
<tr><td valign="top">[a7]</td> <td valign="top"> D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. Thurston, "Wheels, wheeling, and the Kontsevich integral of the unknot" ''preprint'' , '''March''' (1997) pp. q–alg/9703025 {{MR|}} {{ZBL|0964.57010}} </td></tr></table>