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In the mathematical theory of knots, the Kontsevich invariant, also known as the Kontsevich integral, of an oriented framed link is the universal finite type invariant in the sense that any coefficient of the Kontsevich invariant is a finite type invariant, and any finite type invariant can be presented as a linear combination of such coefficients. It was defined by Maxim Kontsevich.

The Kontsevich invariant is a universal quantum invariant in the sense that any quantum invariant may be recovered by substituting the appropriate weight system into any Jacobi diagram.


The Kontsevich invariant is defined by monodromy along solutions of the Knizhnik–Zamolodchikov equations.


Maxim Kontsevich, Vassiliev's knot invariants, Adv. Soviet Math. 16 (1993), 137–150.

Tomotada Ohtsuki, Quantum Invariants – A Study of Knots, 3-Manifolds, and their Sets, Series on Knots and Everything 29, World Scientific Publishing Co., 2002.

Mathematics Encyclopedia

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