Edit: I think LMO is correct. Massuyeau has a nice explanation here.
Edit: Renaud Gauthier has retracted the claim of an error in the foundations of the LMO construction, and has withdrawn both preprints from arXiv.

Original post follows:

In two papers posted to the arXiv in the past few days, Renaud Gauthier claims to have discovered an error in the definition of the framed Kontsevich integral used in the construction of the LMO invariant. I have no reason to doubt him. I looked at these papers some years back and recall that something funny was going on with the normalization under handle-slides. I got the wrong multiple of the normalization factor $\nu$, just as Gauthier does. Gauthier fixes the normalization so that it works, but then remarks that subsequent results depending on this construction need to be carefully checked.

My question is whether anyone knows of results that use the fine details of the definition of the framed Kontsevich integral (or LMO invariant or Aarhus integral) which are now thrown into doubt because of this error.

Edit: Here are links to the papers.

On the foundations of the LMO invariant

On the LMO Invariant, the Wheeling Theorem, and the Aarhus Integral

Edit 2: Moskovich has started a blog post on this. Thanks to Ryan Budney for pointing this out.

A problem with LMO?

  • 1
    $\begingroup$ Add link to the arXiv papers? $\endgroup$
    – Stopple
    Oct 14, 2010 at 16:01
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    $\begingroup$ I don't this post is the place for an exposition of the LMO invariant. Maybe ask as a separate question if you want to know (or read the papers cited the links above!)? $\endgroup$
    – Romeo
    Oct 14, 2010 at 23:20
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    $\begingroup$ Daniel Moskovich just started a blog post about your very question, Jim. ldtopology.wordpress.com/2010/10/14/a-problem-with-lmo $\endgroup$ Oct 15, 2010 at 14:44
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    $\begingroup$ I believe that Gauthier's claim is refuted by Gwenael Massuyeau's comment: ldtopology.wordpress.com/2010/10/14/a-problem-with-lmo $\endgroup$ Oct 20, 2010 at 12:41
  • 2
    $\begingroup$ I am not expert in the details of all of this, but I have noticed one thing as a bystander: Not enough direct communication between Gauthier and the people whose work he criticizes or who criticize his work. His advisor is Soibelman, who is highly respected. Maybe he can help move the debate to a more efficient and more private forum. Because, so far the debate has only been interesting for negative reasons. $\endgroup$ Nov 6, 2010 at 8:21

1 Answer 1


Having been a part of the LMO story from its beginning, and having read and checked all relevant papers carefully at the time, and having taken part in many cross-checks that the LMO invariant passed (normalization-compatibility with Reshetikhin-Turaev, various explicit computations), and having consulted on email with my collaborators at the time, and having superficially read through Gauthier, my informed guess is that in this particular case of inconsistency the first place to look for a problem is in Gauthier, not in LMO.


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