Renaud Gauthier from KSU posted this preprint on ArXiv a few days ago, in which he claims to have found a serious problem with the construction of the LMO invariant, the universal finite-type invariant for rational homology spheres (it’s defined for all 3-manifolds, but I think of it as an invariant of homology spheres). What a headline that makes! A possible hole in the definition of the LMO invariant, with the potential to wash large swaths of quantum topology of 3-manifolds down the drain! Indeed, this is the topic of his second preprint.

Tomotada Ohtsuki was my PhD advisor, and he’s a careful mathematician with tremendous technical ability, who checks his answers against computational data to make absolutely sure no errors creep into his work. Le and Murakami are similar. Gauthier’s claim is that they made a fatal error calculating the effect of the second Kirby move on the framed Kontsevich invariant, which is used to construct the LMO.

Without having read Gauthier’s preprint (which is 81 pages long), my bias is to be skeptical of his claim. Maybe he found a typo, but surely not more. But what a headline it would make if a substantial error was there! This is math drama in the making.

I think it might be fun and educational to crowdsource peer review Gauthier’s claim. It’s important if it’s right, and if it’s wrong, at least it’s an opportunity to get into the kishkes of the LMO.

UPDATE: A MathOverflow question about this.

UPDATE: A third paper by Gauthier was uploaded on Thursday.

UPDATE: Gwenael Massuyeau shows a major flaw in Gauthier’s arguments in the comments. Another two large flaws are noted by Dylan Thurston. Due to these problems, Gauthier’s claim of an error in LMO does not appear to hold up.

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Maybe this isn’t the best location for crowd-sourcing review, but there is already a MathOverflow question about this (though admittedly more about possible consequences rather than assessing the claim).

Unfortunately, I really feel quite unequipped to evaluate the papers. I have to wonder, who did he consult with before putting these papers up? It’s hard for me to imagine that he didn’t really check things extremely carefully before doing this, but it’s also hard to believe such an error would have propagated for so long.

Comment by Ben Webster — October 15, 2010 @ 7:20 pm |

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.

Comment by Dror Bar-Natan — October 16, 2010 @ 5:34 am |

So, he didn’t check with you or any of your collaborators before putting this up? That really seems unwise.

Comment by Ben Webster — October 16, 2010 @ 12:11 pm |

He should have checked with LMO, not with me. I doubt very much that he did, but I have no explicit knowledge.

Comment by Dror Bar-Natan — October 16, 2010 @ 12:30 pm

Sorry, apparently I can’t do 3 replies deep? Anyways, he certainly should have checked with you about the Aarhus integral thing. And if you’re going to post a paper like that, you want to run it past a wide variety of people, so you don’t embarrass yourself to the degree that he will have if he’s wrong.

Comment by Ben Webster — October 16, 2010 @ 3:00 pm

After thinking this over, I have to admit it seems a real stretch that so many good mathematicians would have overlooked this error. In my math overflow post I mentioned that I had gotten the wrong multiple of nu when I did the handle slide, but I think this probably just shows that I made the same mistake as Gauthier!

Comment by James Conant — October 16, 2010 @ 11:19 am |

When I first read the LMO paper, I was also convinced they got the normalization wrong (in just the same way). But I thought about it more, and it is correct, and needs to be the way LMO have it for a variety of reasons, as Dror alluded to.

Gauthier also claims that there are mistakes in degree 1 of the LM paper (see the first paper, arXiv:1010.2422, p. 63), which is just really implausible.

Comment by Dylan Thurston — October 16, 2010 @ 4:03 pm |

There was a factor of two which I found strange at one point, but which Ohtsuki-sensei explained to me. I wonder if it’s the same thing. Anyway, the crux of Gauthier’s argument that there is a problem in Le-Murakami is Proposition 4.1.1 on pages 58-59 and Proposition 4.1.2 on Pages 60-63 of Gauthier. As you point out, Proposition 4.1.2 implies mistakes in degree 1 of Le-Murakami, which seems highly unlikely (although I haven’t carefully gone through Gauthier’s proof). So really, a preliminary impression is that it looks like one has to check (218) and (219) on the bottom of page 59 of Gauthier’s first preprint. I am not following what Gauthier is doing here (although I’ve spent next to no time on it yet). Any ideas?

Comment by dmoskovich — October 16, 2010 @ 9:57 pm

I think I see what Gauthier is doing wrong. He is claiming that when you do a handle slide of a long arc over a trivial unknotted component you get nu^2 on the long arc and nu on the unknotted component. Since such a handle slide is an isotopy, you should just get the standard normalization of nu on the unknot. However, on p.59 of his first paper, to do this he cancels two pairs of critical points, exactly accounting for the extra two factors of nu! See equation (218).

Comment by James Conant — October 17, 2010 @ 5:51 am |

I emailed Gauthier my explanation. We’ll see what he says.

Comment by James Conant — October 17, 2010 @ 6:05 am |

Now I’m confused… (218) is a statement of isotopy invariance, isn’t it?

Comment by dmoskovich — October 17, 2010 @ 8:53 am |

Maybe I was too hasty.

Comment by James Conant — October 19, 2010 @ 7:08 am

Here’s another thing wrong. On the top of page 59 (and in Lemma 3.2.10), he asserts that $\Delta\nu = \nu \otimes \nu$. This is not true, as a short degree-2 computation verifies. This same equation is true if you close off at least one of the components, but that’s not how he’s applying it.

Comment by Dylan Thurston — October 17, 2010 @ 9:10 am |

For the proof of Lemma 3.2.10, both components are closed, which means that there is an extra relation in the space of Jacobi diagrams, which is taking the leg all the way around the closed component of the skeleton. Without this relation, his proof of Lemma 3.2.10 certainly fails. I’m reproducing the degree 2 calculation which you did, to find what the extra contributions look like (and how that effects Gauthier’s argument).

Comment by dmoskovich — October 17, 2010 @ 9:30 am |

I am starting to get worried again. I’m reading Ohtsuki’s book, and it seems to me that Proposition 10.1 does indeed imply that the right hand side of diagram (215) from Gauthier should be the same as the trivial diagram with a copy of \nu labeling the circle component. This isn’t necessarily a contradiction. As Dylan pointed out, the equation Delta(nu)=nu\otimes\nu cannot obviously be applied locally the way Gauthier does it, and indeed applying Lemma 10.2 of Ohtsuki to a q-tangle decomposition of the left-hand side of (215), one does get the diagram on the right. However, I’m trying to understand Ohtsuki’s proof of Lemma 10.2. He argues that equation 10.1 means that you can get rid of the S_1S_2\Delta_3\Phi contribution in the equation directly above, but I don’t see how this follows. Does anyone see the argument?

Comment by James Conant — October 19, 2010 @ 2:02 pm |

I looked at this for a few minutes, walked out of my office, and bumped into Dror Bar-Natan in the common room, who explained it to me with no problem at all. A coproduct of an edge commutes with everything- that’s just a locality result. So you can STU the legs on the fourth skeleton-segment past all the junk, and the associator becomes a sum of chords from the first to the second, and from the first to the first. These commute, and an associator between two commuting variables vanishes. Essentially, that was Ohtsuki’s proof of Lemma 10.2. So there seems to be no problem there.

My feeling is that Dylan’s comment must make everything work- if you were to work out the coproduct of nu correctly, the result would agree with LMO. But I haven’t worked out how yet.

Comment by dmoskovich — October 19, 2010 @ 3:39 pm |

Ah yes. That makes sense. Thanks.

I’m still puzzled though because I don’t think Dylan’s comment saves the day. I agree that the coproduct of nu is not nu\otimes \nu, but the equality of the right-hand side of Gauthier’s (215) and (219) still seems incorrect. Just compare the degree 2 terms. (Alternatively you can force Delta(nu)=nu\otimes nu by dividing the space of diagrams by those diagrams where the two components are connected by at least one chord, and get a contradiction that way.) So something must be going wrong in the application of Ohtsuki’s Prop. 10.1.

When I use Ohtsuki’s prop 10.2 on a q-tangle decomposition of a handle slide over the trivial unknot, I basically get the right-hand side of Gauthier (215), except that the two copies of Delta(nu) are replaced by 4 copies of Delta(nu^(1/2)), which is the same thing. So Ohtsuki’s 10.2 is consistent with 10.1, but 10.1 seems wrong. The conclusion seems to be that I must not be applying 10.1 (and 10.2) correctly, but I don’t see what’s going wrong.

Comment by James Conant — October 19, 2010 @ 6:19 pm

I went directly to the point where Renaud Gauthier claims that the handle-sliding property of the nu-normalized framed Kontsevich integral is not correct, namely Proposition 4.1.1 from his first paper.

One has to be careful with the fact that the band-sum operation is not well-defined at the level of Jacobi diagrams (as it is warmed up, for instance, in Thang Le’s lectures notes from Grenoble 1999) and he seems to apply it in the wrong way, which thus leads to a contradiction. More precisely, it seems to me that he is wrong when he writes at the bottom of page 58: “Now according to (206) and (207) from [O], under a band sum move this should map to (215)”. Here Proposition 10.1 from Ohtsuki’s book [O] does not seem to be applied in the correct way: he’s replacing Delta of the dotted part by \Delta of \nu, but the dotted part is more complicated than just a \nu : two associators should also appear because the two strands which are involved in the band-sum should be “infinitesimally” close to each other. Thus, he is missing a \nu^{-2} factor.

I guess the confusion is coming from the fact that Figure 10.2 in Ohtsuki’s book has to be interpreted in the right way. On the left side of this figure, I see the decomposition of \check{Z}(L) that one gets when L is decomposed into elementary q-tangles *in such a way that* the two vertical strands which we are going to band-sum are parenthesized (..) together. Then \check{Z}(L’) is obtained from this formula for \check{Z}(L) by replacing this box with two vertical strands by a box with one cup, one cap and one vertical strand on the right side parenthesized as (.(..)), by doubling the circle (minus the vertical strand) along which we have slided, and by replacing each univalent vertex which was attached to this circle by a small “box” which grasps the two parallel copies. Then, a “box” is interpreted in the usual way by distributivity (the Delta operation).

Thus, it seems to me that there is no problem in Ohtsuki’s Proposition 10.1.

Comment by Gwenael Massuyeau — October 20, 2010 @ 4:17 am |

And, so far as I can see, this mistake is obviously fatal to Proposition 4.1.1 of Gauthier, the keystone of his claim that LMO is wrong.

And so, this challenge to the correctness of LMO is relegated to the dustbin of history. I can’t say I’m surprised, because LMO has already been checked carefully by many people, and stands as a trusted cornerstone to much other research. But I’m glad to understand these issues now, which are of fundamental importance in the LMO construction.

Comment by dmoskovich — October 20, 2010 @ 7:34 am |

Very nice.

Comment by James Conant — October 20, 2010 @ 7:56 am |