View source for Kontsevich integral

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201001.png" /> (cf. also [[Knot theory|Knot theory]]) embedded in the three-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201002.png" /> in such a way that the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201003.png" /> is a [[Morse function|Morse function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201004.png" /> (all critical points are non-degenerate and all critical levels are different). Its values belong to the graded completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201005.png" /> of the algebra of chord diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201006.png" /> defined below.

The Kontsevich integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201007.png" /> is an iterated integral given by the formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k1201008.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010012.png" />.

2) The integration domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010013.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010014.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010015.png" /> divided by the critical values into a certain number of connected components; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010016.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010017.png" /> intersects the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010018.png" /> in some number, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010019.png" />, of points. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010020.png" /> are constants if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010021.png" /> belongs to a fixed connected component of the integration domain, but in general they can be different for different components. Choose one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010022.png" /> unordered pairs of distinct points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010025.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010026.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010027.png" /> a set of such pairs for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010028.png" />. The integrand is the sum over all choices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010029.png" />.

4) For a pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010030.png" /> the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010031.png" /> denotes the number of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010032.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010034.png" /> where the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010035.png" /> decreases along the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010036.png" />.

5) Fix a pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010037.png" />. Consider the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010038.png" /> as an oriented circle and connect the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010040.png" /> by a chord. One obtains a chord diagram with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010041.png" /> chords. The corresponding element of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010042.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010043.png" />. The algebra of chord diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010044.png" /> is the [[Graded algebra|graded algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010045.png" />. The linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010046.png" /> is generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010047.png" /> by all chord diagrams with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010048.png" /> chords considered modulo the relations of the following two types:

One-term relations:

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

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

Figure: k120100b

for an arbitrary fixed position of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010049.png" /> chords (which are not drawn here) and the two additional chords positioned as shown in the picture.

The multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010050.png" /> is defined by the connected sum of chord diagrams, which is well-defined thanks to the four-term relations. In fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010051.png" /> is even a [[Hopf algebra|Hopf algebra]] but the co-multiplication is not needed here.

6) Over each connected component, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010053.png" /> are smooth functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010054.png" />. With some abuse of notation,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010055.png" /></td> </tr></table>

is to be interpreted as the pullback of this form to the integration domain of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010056.png" />. The integration domain is considered with the positive orientation of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010057.png" /> defined by the natural order of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010058.png" />.

7) By convention, the term in the Kontsevich integral corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010059.png" /> is the (only) chord diagram of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010060.png" /> (without chords) with coefficient one. It represents the unit of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010061.png" />.

The Kontsevich integral is convergent thanks to the one-term relations. It is invariant under deformations of a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010062.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010063.png" />. However, the following formula shows how the integral changes under such deformations:

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

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010064.png" /> in the specified way and the product is the product in the completed algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010065.png" /> of chord diagrams. The last equality allows one to define the universal Vassiliev invariant by the formula

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

Figure: k120100d

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010066.png" /> denotes the number of critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010067.png" /> and the quotient means division in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010068.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010069.png" /></td> </tr></table>

The universal Vassiliev invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010070.png" /> is invariant under an arbitrary deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010071.png" />.

Consider a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010072.png" /> on the set of chord diagrams with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010073.png" /> chords satisfying one- and four-term relations. Applying this function to the universal Vassiliev invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010074.png" />, one obtains a numerical knot invariant. This invariant will be a Vassiliev invariant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010075.png" /> 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 style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k120100e.gif" />

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010076.png" />). The claim that the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120100/k12010077.png" /> 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</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</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)</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</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</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</TD></TR></table>