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• CommentRowNumber1.
• CommentAuthorRichard Williamson
• CommentTimeMay 1st 2018
• (edited Jun 8th 2018)

I would like to make a new attempt to describe my work on an approach to the Poincaré conjecture for knots. The arguments have essentially been stable for a couple of years, though I have a tidied up a few details here and there. See the nLab page Poincaré conjecture - diagrammatic formulation for the reformulation of the Poincaré conjecture into a statement in diagrammatic theory. It is this reformulated statement that I will discuss here; everything at the nLab page involved in the reformulation can be regarded as a black box.

Begin, then, with a knot diagram $K$. Fix a point $p$ on it. Equip it with an orientation (any will do), and use this orientation to define its fundamental group and longitude. Label the arcs of $K$. By a word in the arcs of $K$, I shall mean a monomial $a_{1}^{\pm 1} \cdots a_{n}^{\pm 1}$, where $a_{1}$, $\ldots$, $a_{n}$ are (labels of) arcs of $K$. I will say that a word $w$ in the arcs of $K$ is realisable if we can find a welded knot $K_{w}$ which is equivalent to $K$ under the welded framed Reidemeister moves (ordinary framed Reidemeister moves plus the virtual Reidemeister moves and one of the forbidden moves, namely that which allows to slide a classical arc over a virtual crossing), and which has the following property.

First, if there are any occurrences of the longitude $l$ of $K$ as sub-words of $w$, then remove them all except one (any of the ways to do this may be used). For ease of notation, I will suppose that we have already done this for $w$. Secondly, we allow that we apply a permutation to $w$. Again, I will assume that we have already done this. Finally, we allow that if, for some arc $a$ of $K$, both $a$ and $a^{-1}$ occur in $w$ (not necessarily consecutively), then we can remove them from $w$. Once again, I will assume that we have made any such deletions that we wish to make. Then we ask that the following is the case.

1) There is a point $q$ on $K_{w}$ such that as we walk around $K_{w}$ exactly once, in the direction defined by the orientation, beginning at $q$ and returning to $q$, then we pass successively, in order (though this doesn’t really matter, since we allow a permutation to be applied to $w$ as a preliminary step), and ignoring when we pass through virtual crossings and over a crossing, under the arcs $a_{1}$, \ldots, $a_{n}$, and under no other arcs.

2) The power of $a_{i}$ is the sign of the crossing at which we pass under $a_{i}$.

Given a crossing $C$ of $K$ as follows, irrespective of the orientation of the horizontal arcs, I shall denote by $w_{C}$ the word $c^{-1}b^{-1}ab$ in the arcs of $K$.

     / \
|
|
----- | ----
c   |   a
|
b


Now, $\pi_{1}(K) / \langle l \rangle$ is isomorphic to the quotient of the free group $F(K)$ on the arcs of $K$ by the normal subgroup $N$ consisting exactly of words in the arcs of $K$ of the form $a^{-1}va$ and their inverses, where $v$ is any concatenation of copies of $l$ and of words in the arcs of $K$ of the form $w_{C}$ for various crossings $C$ of $K$, and where $a$ is any arc of $K$.

My key claim is that every word of $N$ of the form $g^{-1}vg$ which contains a copy of $l$ is realisable, and moreover, if a word $w$ is equivalent to such a word of $N$ under the equivalence relation of being able to add or delete pairs $aa^{-1}$ and $a^{-1}a$, where $a$ is an arc of $K$, then $w$ is also realisable.

Let us prove this. Take any $v$ as above. Since we delete all copies of $l$ from $v$ except one when defining realisability, and since I am assuming that there is at least one copy of $l$, we may assume that $v$ is of the form $w_{C_{1}} \cdots w_{C_{i}} \cdot l \cdot w_{C_{i+1}} \cdots w_{C_{n}}$ for some $n$.

We begin at $p$. We will walk around $K$ in the same direction as our original orientation which we are using to define $\pi_{1}(K)$ and $l$. Suppose that $w_{C_{1}}$ looks as in the figure above. Take a small piece of the arc on which $p$ lies, just after $p$. Drag it, using only virtual R2 moves, so that it is near the above figure. We suppose that the arc on which $p$ lies has label $d$.

            / \
|
|
|
|
<----------- | ----------
c      |     a
|
-----    |
|     |   |
d |     |   |
|    \ /  |
b


Then slide it (using two R2 moves and an R3 move) under the above crossing, so that we have the following local picture. Noam Zeilberger described this move aptly as a ’lasso move’ in an earlier discussion.

            / \
|
------- | ---
|        |    |
|        |    |
<----------- | ----------
|   c    |    |  a
d |        |    |
<-- | ---
d   |
|
b


Note that we do not label the new arcs, i.e. we leave the labellings as they were except that there is a ’break’ in the arc labelled $d$. An entirely analogous construction can be given in the case that the horizontal arcs have the opposite orientation.

We now proceed in exactly the same way for $w_{C_{2}}$, using the arc labelled $d$ with an arrow on it in the above figure. And so on until we have done the same for $w_{C_{i}}$. At this point, we now encounter $l$ in our word. And we now walk all the way around the virtual knot which we have obtained so far, from where we were after carrying out the above procedure for $w_{C_{i}}$, stopping when we reach arc $d$, a little before we reach $p$. After this, we carry out the procedure above for $w_{C_{i+1}}$, …, $w_{C_{n}}$, beginning with a little piece of arc between where we stopped and $p$. After we are finished with $w_{C_{n}}$, we simply walk to $p$.

This completes the realisation of $v$.

Now, take the welded knot $K_{v}$ that we have constructed to realise $v$. To realise $a^{-1}va$, where $a$ is any arc of $K$, we proceed as follows. Take a small piece of the arc $a$. Using virtual R2 moves, drag it across $K_{v}$ so that it is near the point $p$. Then apply an R2 move so that we have the following local picture.

     ---------
|         |
--- | ---•--- | --->
d   |    p    |
|        \ /
a


Walking around our new virtual knot from $p$ in the same direction as before, we obtain a realisation of $a^{-1}va$.

it remains to show that if we add or delete a pair $bb^{-1}$ or $b^{-1}b$, where $b$ is an arc of $K$, from such a $a^{-1}va$, then the resulting word is also realisable. To add a pair $bb^{-1}$ between, say, $x$ and $y$ in $a^{-1}va$, then we apply the same idea that we have just seen: take a small piece of the arc labelled $b$ on $K_{a^{-1}va}$, drag it using virtual R2 moves so that it is near $x$ and $y$, and then apply an R2 move so that we have the following picture.

   / \     -----     / \
|     |     |     |
--- | --- | --- | --- | ---
|     |     |     |
|    \ /    |     |
x           b     y


The same argument works for adding $b^{-1}b$, just using the virtual R2 moves in a different way so that we can drag the arc $b$ over from the opposite side; and of course we could have $x^{-1}$ or $y^{-1}$ or both, and would be able to apply the same argument.

Suppose now that we wish to delete a pair $bb^{-1}$ from $K_{a^{-1}va}$. This is a crucial part of the argument, and we need to be careful. The idea is simple. If we have a pair $bb^{-1}$, then, ignoring virtual crossings and crossings which we travel over, we must successively walk under $a$ and then under $a$ again in the opposite direction, without walking under any other arcs. This means that we have a local picture as follows, except that there may be other arcs which pass under those shown, or which cross those shown vertically.

     -----
|     |
--- | --- | ---
|     |
|    \ /
b


Now, in welded knot theory, even if there are arcs which cross under those shown or cross them vertically, we can slide the arc $a$ over the other depicted arc, so that we have the following local picture.

---------------

-----
|     |
|     |
|     |
|    \ /
b


With only virtual moves available, i.e. without the forbidden move, we would not necessarily be able to carry out this slide, because we would not be able to handle the case that there were some virtual crossings involved. This is the reason that we work in welded knot theory and not virtual knot theory.

A second point of note which we must take care to address is that this slide may permute the arcs making up $a^{-1}va$, and may lead to further deletions of $b$ and $b^{-1}$ (always in pairs, though this $b$ and $b^{-1}$ might not occur consecutively) from $a^{-1}va$. But both of these operations are permitted in the definition of realisability (they are two of the three preliminary operations that may be applied).

The same argument works for deleting a pair $b^{-1}b$.

This completes the demonstration of the claim. Suppose that $\pi_{1}(K) / \langle l \rangle$ is trivial, so that $N$ is $F(K)$.

Let $b$ be any arc of $K$. Since $N$ is all of $F(K)$, either $b$ or $b^{-1}$ is equal in $F(K)$ to a word of the form $a^{-1}va$ for some $a$ and $v$, where $v$ is any concatenation of copies of $l$ and of words in the arcs of $K$ of the form $w_{C}$ for some crossings $C$ of $K$.

I now claim that for at least one arc $b$ of $K$, the word $v$ in the word $a^{-1}va$ which is equal to either $b$ or $b^{-1}$ contains at least one copy of $l$. Indeed, if this is not the case, then every arc of $K$ or its inverse is equal in $N$ to a $a^{-1}va$ where $v$ is a product of $w_{C}$’s. That is to say, every arc of $K$ or its inverse is then trivial in $\pi_{1}(K)$. But this implies that $\pi_{1}(K)$ is trivial, and no knot has a trivial fundamental group.

Putting the two claims together, we obtain that, for at least one arc $b$ of $K$, the word consisting just of $b$ or of $b^{-1}$ is realisable. This could happen in two ways. One is that $b$ or $b^{-1}$ is the longitude of $K$. In this case, $K$ must be a $\pm 1$-framed unknot.

The other is that there is a virtual knot $K_{b}$ which is equivalent to $K$ as a framed welded knot, and on which there is a point $q$ from which, when we walk around $K_{b}$ in a particular direction and return back to $q$, we pass only under a single arc, namely $b$ (note that no non-trivial permutations and deletions are possible in this case).

But every welded knot with only one classical crossing is equivalent (as a welded knot) to a classical $\pm 1$-framed unknot. Hence $K$ itself is equivalent as a welded knot to a $\pm 1$-framed unknot. But a pair of classical knots are equivalent as welded knots if and only if they are equivalent as classical knots. We conclude that $K$ is equivalent to a $\pm 1$-framed unknot as a classical knot.

If this is correct, this is the Poincaré conjecture for knots (also known as the Property P conjecture). In fact, since we do not use any Kirby moves, the result is a bit stronger: I believe that it also establishes the Gordon-Luecke theorem (if a Dehn surgery on a knot gives $S^{3}$, then the knot must be the unknot).

I believe that the argument adapts to arbitrary links, with one caveat: we do need Kirby moves in that case, and we then need a version of the fact that classical links are welded-equivalent if and only if they are classically equivalent which includes certain kinds of Kirby moves. This is conjectured in the literature, but has not previously been known. I had tried off and on for a year or more to prove it without succeeding (I am trying to avoid the use of Waldhausen’s work on 3-manifolds), but I now do think to have a proof, using not exactly the same moves as in welded knot theory (for these moves I still do not have a proof), but a modified collection for which the arguments still go through. But let’s just ignore the links case, for now, I suggest; I am very happy to provide details to anybody interested.

I would be delighted with any feedback. As I say, the argument has been in a stable form for getting on for two years, and I’d really like to find out whether or not this argument is essentially correct, even if it turns out to be nonsense or crucially flawed. Since I seem to have immense difficulty in writing it down in a paper, this forum may be the best place for me to communicate the argument. I am very happy to elaborate on anything or to give examples.

1. I am wondering about the possibility of submitting this as a ’Publications of the nLab’ article. Does anybody have any thoughts on that? I.e. is the ’publications of the nLab’ project still in principle active?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeMay 4th 2018

It hardly got going, with only two publications there. I doubt the ’tentatively confirmed’ names on the editoral board still consider themselves in that light.

2. Indeed! I really like many ideas of the project, though, especially the hyperlinking and the transparency of the review process (one could even imagine taking it one step further, where there is an open discussion at the nForum). I suppose the editorial board would probably not be appropriate for the stuff in #1 anyhow.

3. I am thinking of trying to build up a write up of the above on the nLab, which I then produce a pdf from. Would this be better in a personal web, or would it be OK to use the main nLab, as long as the pages are indicated to be unpublished research? I’d incline to the second option myself, but am happy to follow whatever the consensus is.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeAug 26th 2018

Personally, I think it makes sense to keep dedicated pages on a personal web (perhaps read-only for others), so that you could preserve your personal vision as well as maintain a central repository for this project. There is always a possibility of copying over to main those parts of the project that have matured under the eyes of yourself and perhaps more especially others who are following what you are doing, and that would be sensibly assimilated into the main lab. (For example, any improvements in the exposition of known results would of course be very welcome.) You could also link, within articles on main, back to your personal pages.

I seem to recall a few years ago that Emily Riehl had put some critical questions to you over the earlier version of the project, but don’t know what became of that. I believe you were in contact with Lou Kauffman as well, but I don’t think I ever saw anything of his own reactions. Was or is anyone else following your more recent developments?

As you know, Urs maintains a lot of his projects on his personal web, so he might be able to describe in more detail how the dialectic between his web and the main lab works out in practice.

I wish you luck with this project. Far be it from me to give advice, but obviously the experts will want to know what are the main new ideas which sets this apart from other attempts which bear a family resemblance to this (I seem to recall you anticipated this to some extent in the earlier version, but don’t recall what you said specifically, beyond a plea to put preconceptions aside).

• CommentRowNumber7.
• CommentAuthorRichard Williamson
• CommentTimeAug 26th 2018
• (edited Aug 26th 2018)

Thanks very much for the reply, Todd! Before I received your message, a few hours ago, I made a start at towards a diagrammatic proof of the Poincaré conjecture for knots at the main nLab. But I can easily move this to a personal web if that is preferred (and if people do not mind me having a personal web, I do not have one at the moment), it would be absolutely fine by me. I am trying to use the work as a point of departure for improving the existing nLab pages on knot theory (have edited a couple today), and for adding some new ones.

I seem to recall a few years ago that Emily Riehl had put some critical questions to you over the earlier version of the project, but don’t know what became of that. I believe you were in contact with Lou Kauffman as well, but I don’t think I ever saw anything of his own reactions. Was or is anyone else following your more recent developments?

Nobody unfortunately has offered any reactions on recent versions. Lou Kauffman is kindly still interested, but I have not received any mathematical feedback. The project has moved on quite a bit from the earlier time you are referring to; the fundamental ideas have been the same all the way through, but a few technical points have been tidied up, and a few new ideas occurred along the way which helped to make it easier to write up. The current version has been stable for quite a long time, though, over 1.5 years in the knot case. I have tried to ask people to look at it, but with no success. I think the only thing I can do is submit it to a journal, which is what I plan to do (write it up on the nLab and export to a pdf).

As you know, Urs maintains a lot of his projects on his personal web, so he might be able to describe in more detail how the dialectic between his web and the main lab works out in practice.

Absolutely, I’d be very happy to hear from Urs about what he suggests.

I wish you luck with this project.

Thank you very much, I appreciate it!

Far be it from me to give advice, but obviously the experts will want to know what are the main new ideas which sets this apart from other attempts which bear a family resemblance to this (I seem to recall you anticipated this to some extent in the earlier version, but don’t recall what you said specifically, beyond a plea to put preconceptions aside).

Yes, definitely. I feel able to answer this. Indeed, the approach is completely different to anything I’ve seen, to the extent that one notable expert (whose name has not been mentioned in any of my public posts on this work) was not able to be convinced of the validity of the logic of the approach, which I don’t think is in any doubt. The use of virtual knot theory is for instance a novel technical aspect, as far as I know. But I’ll treat this in more detail in the introduction, when I get to it.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeAug 26th 2018

I second Todd, this would be good to do on a private web. Also, having a private web would be useful for you, as the developer/maintainer of the main lab, to try out some things on a decent collection of mathematical material without nudging the nLab too much.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeAug 26th 2018

I can’t speak for others on the Steering Committee, but as a member I would definitely support the SC granting you a personal web (which sounds funny, as I guess you might be the one setting it up!).

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeAug 27th 2018

Yes, I agree with what was said above by Todd and David R.

4. OK, thanks all! I will move to a personal web later.