The Kontsevich integral is known to be a universal Vassiliev invariant. It is still an open question whether it is a complete knot invariant, i.e. whether it distinguishes a given knot from all other knots up to ambient isotopy and mirror image. I am curious to know if there are some references giving the state of the art for this important question. Have some clues for a possible answer emerged in the last few years?

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    $\begingroup$ Certainly there are clues. My feeling is Vassiliev invariants do not distinguish knots. The universal (integral) finite-type invariant is describable via the Taylor Tower for the Goodwillie-Weiss embedding calculus. It appears this tower must also have a description in terms of operads. I think once this is worked out your question will be resolved. $\endgroup$ Oct 11, 2013 at 10:33
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    $\begingroup$ If one restricts to hyperbolic knots, then a positive answer to the volume conjecture would imply that there are at most finitely many hyperbolic knots with the same Kontsevich integral. I'm not expert enough to say whether there has been much progress on the volume conjecture though. $\endgroup$
    – Ian Agol
    Oct 11, 2013 at 16:06

1 Answer 1


I don't think that there has been a tremendous amount of progress in understanding the Kontsevich Invariant of a knot in the last decade or so. It appears that essential new ideas may be needed in order to answer the fundamental questions.

Quantum Invariants: A study of knots, $3$-manifolds, and their sets, T. Ohtsuki, World Scientific 2002.
This is still more-or-less current, I think, and is a basic reference. Also T. Ohtsuki's Problems on invariants of knots and 3–manifolds.

A more recent textbook reference is:

Introduction to the Vassiliev Knot Invariants by Chmutov, Duzhin, and Mostovoy, Cambridge University Press, 2012.
It contains a detailed discussion of the Kontsevich Invariant of a knot, including developments of the last decade such as work that has been done on the loop expansion and on the 2-loop polynomial in particular.

With regard to clues for a final answer, not to toot my own horn, but Ohtsuki and I showed that 3 loop Vassiliev invariants do not distinguish a knot from its reverse (this is trivial for lower loop degree):

Vanishing of 3-loop Jacobi diagrams of odd degree, D. Moskovich and T. Ohtsuki, J. Comb. Theory, Ser. A 114(5): 919-930 (2007).
If this were true for general loop degree, then that would imply that there exist prime unoriented knots which cannot be distinguished by the Kontsevich invariant (this is a result of Kuperberg). My personal suspicion (without solid support) is that this is false for general loop degree, and indeed that the Kontsevich invariant does separate (unoriented) knot types.

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