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Kontsevich Integral


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 R^3 as a direct product of a complex line C with coordinate z and a real line R with coordinate t. The integral is defined for Morse knots, i.e., knots K embedded in R^3=C_z×R_t in such a way that the coordinate t is a Morse function on K, and its values belong to the graded completion h^_calA of the algebra of chord diagrams A.

The Kontsevich integral Z(K) of the knot K is defined as

 Z(K)=sum_(m=0)^infty1/((2pii)^m)int_(t_(min)<t_1<...<t_m<t_(max); t_j are noncritical)sum_(P={(z_j,z_j^')})(-1)^vD_P ^ _(j=1)^m(dz_j-dz_j^')/(z_j-z_j^'),
(1)

where the ingredients of this formula have the following meanings. The real numbers t_(min) and t_(max) are the minimum and the maximum of the function t on K.

KontsevichIntegral

The integration domain is the m-dimensional simplex t_(min)<t_1<...<t_m<t_(max) divided by the critical values into a certain number of connected components. For example, for the embedding of the unknot and m=2 (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 {t=t_j} subset R^3, choose an unordered pair of distinct points (z_j,t_j) and (z_j^',t_j) on K so that z_j(t_j) and z_j^'(t_j) are continuous functions. Denote by P={(z_j,z_j^')} the set of such pairs for j=1, ..., m, then the integrand is the sum over all choices of P. In the example above, for the component {t_(min)<t_1<t_(c_1),t_(c_2)<t_2<t_(max)}, we have only one possible pair of points on the levels {t=t_1} and {t=t_2}. Therefore, the sum over P for this component consists of only one addend. In contrast, in the component {t_(min)<t_1<t_(c_1),t_(c_1)<t_2<t_(c_2)}, we still have only one possibility for the level {t=t_1}, but the plane {t=t_2} intersects our knot K in four points. So we have (4; 2)=6 possible pairs (z_2,z_2^'), and the total number of addends is six (see the picture below).

For a pairing P the symbol 'v' denotes the number of points (z_j,t_j) or (z_j^',t_j) in P where the coordinate t decreases along the orientation of K.

KontsevichChordDiagram

Fix a pairing P, consider the knot K as an oriented circle, and connect the points (z_j,t_j) and (z_j^',t_j) by a chord to obtain a chord diagram with m chords. The corresponding element of the algebra A is denoted D_P. In the picture above, one of the possible pairings, the corresponding chord diagram with the sign (-1)^v, and the number of addends of the integrand (some of which are equal to zero in A due to a one-term relation) are shown for each connected component.

Over each connected component, z_j and z_j^' are smooth functions in t_j. By  ^ _(j=1)^m(dz_j-dz_j^')/(z_j-z_j^') we mean the pullback of this form to the integration domain of variables t_1, ..., t_m. The integration domain is considered with the manifold orientation of the space R^m defined by the natural order of the coordinates t_1, ..., t_m.

By convention, the term in the Kontsevich integral corresponding to m=0 is the (only) chord diagram of order 0 with coefficient one. It represents the unit of the algebra A.

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 t. However, the formula shows how the integral changes under such deformations:

KontsevichDeformation

In the above equation, the graphical arguments of Z represent two embeddings of an arbitrary knot, differing only in the illustrated fragment,

KontsevichHump

H is the hump (i.e., the unknot embedded in R^3 in the specified way; illustrated above), and the product is the product in the completed algebra h^_calA of chord diagrams. The last equality allows the definition of the universal Vassiliev invariant by the formula

 I(K)=(Z(K))/(Z(H)^(c/2)),
(2)

where c denotes the number of critical points of K and quotient means division in the algebra h^_calA according to the rule (1+a)^(-1)=1-a+a^2-a^3+.... The universal Vassiliev invariant I(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 (a weight system). Applying this function to the universal Vassiliev invariant w(I(K)), we get 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 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:

 I^'(K_1#K_2)=I^'(K_1)I^'(K_2),
(3)

where I^'(K)=Z(H)I(K). For any knot K the coefficients in the expansion of Z(K) 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 I(K) is currently known only for the unknot,

I(O)=exp(sum_(n=0)^(infty)b_(2n)w_(2n))
(4)
=1+(sum_(n=0)^(infty)b_(2n)w_(2n))+1/2(sum_(n=0)^(infty)b_(2n)w_(2n))^2+....
(5)

(Bar-Natan et al. 1995). Here, b_(2n) are modified Bernoulli numbers, i.e., the coefficients of the Taylor series

 sum_(n=0)^inftyb_(2n)x^(2n)=1/2ln((e^(x/2)-e^(-x/2))/(1/2x))
(6)

(b_2=1/48, b_4=-1/5760, ...; OEIS A057868), and w_(2n) are the wheels, i.e., diagrams of the form

KontsevichWheels

The linear combination is understood as an element of the algebra of Chinese characters B, which is isomorphic to the algebra of chord diagrams A. Expressed through chord diagrams, the beginning of this series looks as follows:

KontsevichSeries

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).


See also

Chord Diagram, Gauss Integral, Morse Knot, Vassiliev Invariant

This entry contributed by Sergei Duzhin

This entry contributed by Sergei Chmutov

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References

Arnol'd, V. I. "Vassiliev's Theory of Discriminants and Knots." In First European Congress of Mathematics, Vol. 1 (Paris, 1992) (Ed. A. Joseph, F. Mignot, F. Murat, B. Prum, and R. Rentschler). Basel, Switzerland: Birkhäuser, pp. 3-29, 1994.Bar-Natan, D. "On the Vassiliev Knot Invariants." Topology 34, 423-472, 1995.Bar-Natan, D.; Garoufalidis, S.; Rozansky, L.; and Thurston, D. "Wheels, Wheeling, and the Kontsevich Integral of the Unknot." Israel J. Math. 119, 217-237, 2000.Chmutov, S. V. and Duzhin, S. V. "The Kontsevich Integral." Acta Appl. Math. 66, 155-190, 2000.Kontsevich, M. "Vassiliev's Knot Invariants." Adv. Soviet Math. 16, Part 2, 137-150, 1993.Le, T. Q. T. and Murakami, J. "The Universal Vassiliev-Kontsevich Invariant for Framed Oriented Links." Compos. Math. 102, 42-64, 1996.Sloane, N. J. A. Sequence A057868 in "The On-Line Encyclopedia of Integer Sequences."Vassiliev, V. A. "Cohomology of Knot Spaces." In Theory of Singularities and Its Applications (Ed. V. I. Arnold). Adv. Soviet Math. 1, 23-69, 1990.

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Kontsevich Integral

Cite this as:

Chmutov, Sergei and Duzhin, Sergei. "Kontsevich Integral." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/KontsevichIntegral.html

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