Update: Renaud Gauthier has retracted the claim of an error in the foundations of the LMO construction, and has withdrawn both preprints from arXiv.
A few weeks ago, we discussed a preprint by Renaud Gauthier which claims a fatal error in the construction of the LMO invariant, having to do with invariance under the Kirby II move of a certain renormalization of the framed Kontsevich invariant which is used to construct the LMO. It also claims to correct that error with an alternative normalization- I do not discuss that claim. In this post, I would like to take stock, to explain Gauthier’s claim and Massuyeau’s response, and to tell you where I think we stand now.
The foundation of 3-manifold quantum topology is Edward Witten’s Quantum Field Theory and the Jones Polynomial, in which he constructs a Jones polynomial for 3-manifolds at the physical level of rigour. A whole mini-discipline of mathematics sprang up around understanding Witten’s invariant mathematically, and we’re still a long way from achieving that goal. If we restrict and restrict and restrict, until we only consider integral homology 3-spheres and the contribution of the trivial flat connection to Witten’s invariant, then (modulo various minor issues) Witten’s invariant becomes (equivalent to) something called the Reshetikhin-Turaev invariant.
These invariants- Witten’s invariant, the Reshetikhin-Turaev invariant, and indeed the LMO invariant- are built by exploiting the fact that a 3-manifold can be represented via Dehn surgery as a framed link in . Topologically, this works by considering with the framed link sitting in , and attaching a 2-handle to each link component as specified by the framing. The top boundary of the space which you end up with is your 3-manifold . For intuition, think one dimension down, where you would be attaching 1-handles to pairs of points on the top boundary of “disk cross unit interval” to obtain some surface as the new top-boundary.
Kirby’s theorem tells us that two framed links present the same 3-manifold only if they are related by blow-ups and by Kirby II (handleslides):
The recipe for constructing a 3-manifold invariant from a surgery presentation (i.e. from a framed link) is to take your favourite link invariant, to extend it to a framed link invariant, and to mod out its values by relations induced by the Kirby moves. The best case is when your framed link invariant is invariant under Kirby moves. Quantum invariants are local (recall as we discussed here that this is perhaps their defining property), so the above recipe can be carried out explicitly.
The universal finite-type invariant for links is the Kontsevich invariant. You can extend it to a framed link invariant . There’s a standard way to make it invariant under Kirby I. But- and this is the key point- is not invariant under Kirby II. A key insight of Le, Hitoshi and Jun Murakami, and Ohtsuki (I don’t know whose insight it was first… but one of these people) is that DOES become invariant under Kirby II if you renormalize by summing in a certain element for each component of , which converts into a new framed link invariant . Everything works out wonderfully and you get the LMO invariant, which is the universal quantum invariant for 3-manifolds. It fits in well with 3-manifold topology: it recovers the Reshetikhin-Turaev invariant and the Alexander polynomial, and its degree 1 part recovers the Casson invariant.
Renaud Gauthier claimed that the normalization of is wrong, and that it is not invariant under Kirby II moves. The claim was then that each component should have been summed in with an element instead of with ; and that doing so makes the framed Kontsevich invariant of invariant under Kirby II.
There’s something which we have to understand about Kirby II before evaluating this claim: namely that the handleslide is not a well-defined move between links. Consider two link components which are a long way apart, and between are lions, tigers, and bears. To slide over , first a part of it must be brought near a part of , and there are many ways of doing this. Two different approach paths may lead to two different links. Consider:
To upgrade Kirby II to a well-defined move on links, which is a prerequisite for calculating the correct normalization, a path between a point in and a point in must be specified. LMO do this by equipping and with points which are infinitesmally close together. Gauthier doesn’t do this, and he arrives at different results. Observe:
Massuyeau said this in a comment to the last post; and it accounts for the disrepancy between Gauthier’s results in Section 4, and LMO.
This is unsurprising given that, computationally, the LMO construction is compatible with the Reshetikhin-Turaev invariant, and therefore with Witten’s invariant. It would shake our very view of physics if it were to contain a substantial error.
The end of the story?
Despite the apparent failure of Gauthier’s claim of an LMO error, this affair does raise a number of important and disturbing points:
- The language of LMO is very confusing. The calculations always seem to hang by a narrow thread, and the fact that everything cancels out at the end seems little short of a miracle. A better language is sorely needed; and LMO needs to be rewritten in that language. Zsuzsana Dancso is doing this (at least for LMMO).
- I don’t know a conceptual reason why needs to be renormalized, and why the renormalization factor has to be for each link component (as opposed to or to something entirely different). Thus our confidence that the LMO normalization is correct relies on computational verification as opposed to on understanding. This is unsatisfactory.
- Perhaps the LMO normalization is not unique? It is a-priori possible that many different renormalizations of the framed Kontsevich invariant could give rise to different 3-manifold invariants. None of these (except for ) wod be compatible with physics, but so what? It doesn’t mean they can’t exist!
These are the important foundational questions which Gauthier’s preprint raises. They can no longer be ignored.
EDIT: See also Dror’s comments on this post.