My feeling is single triangulations or gems is not the right setting. Much like how solutions to the gluing equations for hyperbolic structures depend on the precise triangulation you use, a combinatorial Ricci flow would probably have to live on a space of triangulations of the manifold.

]]>Dear Ryan,

I hope that it is cIear that am not trying to find flaws in SnapPy. I think is a genuine scientific question to test its isomorphic triangulation algorithm with mine which is based in TS-moves. This last one has been very strong in proving when two gems

induce the same 3-manifold.

After Perelman I am very interested in seeing whether we can find polynomial algorithms to describe the structure of the 3-manifolds solely from the combinatorics of the gem. I think, given our current knowledge, this became a feasible project for closed 3-manifolds.The issue is that the structure appears at the level of decomposing the gem. I have a lot to learn from you

in the topic.

So although I doubt anyone would assert SnapPy is flawless or that it attempts to do everything well, it’s probably one of the most looked-over pieces of software in low-dimensional topology, and has withstood many tests, as many people have used it for many different purposes.

I’m not sure to what extent topologists use SnapPy to solve the homeomorphism problem. I think it’s more often used to study properties of particular manifolds. For a long time my primary interest in SnapPy was its ability to compute isometry groups. I use it fairly systematically now in a partial-algorithm for computing the geometric decomposition of manifolds — a solution to the homeomorphism problem is not enough for my purposes. The manifolds that come up in my application typically have connect-sum and JSJ-decompositions, so SnapPy is just a step in a procedure to find the geometric “name” of the manifold.

]]>“But it is beyond ridiculous to state that “the onus is on SnapPy to reproduce” anything, including the BLINK tabulation. And if you feel there are hundreds of thousands of tests to be made, then I suggest that you get started making them.¨

I was surprise and did not understand what seemed to me an over reaction of Marc on my last comments. The reason, I realize later, is that we were talking about two different issues.

First let me say of my great admiration for the program SnapPy. To the list of people that he brings forth having computational projects on 3-manifolds which were helped by SnapPy he should append our names. By using SnapPy the complete topological classification of U9 was completed. And that of T16 is under its way.

Marc’s words:

“I think it has been amply demonstrated on this blog that SnapPy would be useful for your tabulation project as well. When it is feasible, and we think it will be useful, we try to incorporate the data produced by such projects as part of SnapPy’s database of manifolds. You are more than welcome to use SnapPy in your project, and If you would like for us to include your BLINK tabulation in SnapPy, we can discuss that in private.”

I rush to say that I couldn’t agree more with you Marc and that I happily accept all the 3 invitations contained in the above paragraph. I also want to thank you for being so helpful in guiding Cristiana to install SnapPy in our machines.

Now for removal of the misunderstanding. There are two types of onuses at stake. I will classify them as psychological and epistemological onuses. In the first case I completely agree with Marc in his reaction. The point is that I meant, and this was not clear, the epistemological.

To make my point clear I abstract the situation in order to take distance form the emotional aspects of the discussion.

Suppose we have a system X operating on an infinite set O

which needs to be partitioned into equivalence classes.

X has partial solution for the two problems on a pair (o,o’) of members of O: sometimes it can prove that o and o’ are equivalent, sometimes it can show that they are not equivalent.

Sometimes it fails: the pair (o,o’) remains in doubt.

Now we have another system, named Y, operating on a subset U of O which does the following on the same equivalence classes as X restricted to U. First U is formed by larger and larger finite sets U’ so that at a fixed instant in time Y has achieved the following conditions: for any given pair in (u,u’) in U’, either Y produces a proof that u,u’ are equivalent or else it shows that they are not equivalent. Proof here has the following formal connotation: it is an output, that other systems, including X, can consult to convince itself that Y is right about (u,u’).

Those are my grounding setting premisses. Now it follows, almost by definition that for each pair (u,u’) declared equivalent by Y, X has the epistemological onus of checking whether it also can prove them to be equivalent. And if it fails to do so on any single pair, this means under the ground rules that X is not complete and that, in the name of truth, can profit from the situation by including some aspects of Y in its internal design.

Note that this does not mean that the developers of X must stop doing what they are doing to reproduce Y’s computations. Of course, the psicological onus is on the developers of Y, to do the work and so prove that Y is not subsumed by X. Particularizing to our situation, I was not suggesting to the SnapPy developers ought to check the hundreds of thousands of examples that should be tested into X. This work is to be performed by the developers of Y. Only they have the incentive to do it. I will seek help for doing, have no doubt about this.

I devised this ideal test when I was challenged by Henry Wilton to produce with BLINK something that SnapPy could not produce. The problem is that I did not explain myself explicitly and this resulted in this misunderstanding. Let us move on and collaborate in the name of progress of the science we all love: the mysterious 3-manifolds.

The internal algorithms for proving homeomorphisms of BLINK and of SnapPy are based on different techniques and on the different types of triangulations. I think, in the name of truth, that it is an interesting project to find more about these differences and include both algorithms in the same package, if we find it to be worth. On this matter I have the feeling that they are the algorithms are incomparable (none is stronger than the other) and so, the integration will be an advance.

]]>Thanks Ryan – that’s more than enough detail! Anyway, I was being silly above when I suggested that I only knew how to rigorously enumerate hyperbolic 3-manifolds using Manning’s algorithm. Of course one can take the complementary route, as you suggest – use normal surface theory to find essential tori (plus deal with the other atoroidal manifolds somehow) and the remaining 3-manifolds are hyperbolic, by geometrization.

]]>“I think the onus is on SnapPy to reproduce the BLINK …”

Sóstenes, I couldn’t disagree more.

SnapPy is a tool for studying hyperbolic 3-manifolds. It is available to anyone who wants to use it. It can be useful for tabulating links and closed manifolds. In fact, the SnapPea kernel has been used in a number of tabulation projects by a number of people, including, in various combinations, Mark Bell, Pat Callahan, Abhijit Champanernaker, Nathan Dunfield, Stavros Garoufalidis, Matthias Goerner, Martin Hildebrand, Craig Hodgsen, Jim Hoste, Ilya Kofman, Eric Patterson, Saul Schleimer, Morwen Thistlethwaite, Jeff Weeks and others. I think it has been amply demonstrated on this blog that SnapPy would be useful for your tabulation project as well. When it is feasible, and we think it will be useful, we try to incorporate the data produced by such projects as part of SnapPy’s database of manifolds. You are more than welcome to use SnapPy in your project, and If you would like for us to include your BLINK tabulation in SnapPy, we can discuss that in private.

But it is beyond ridiculous to state that “the onus is on SnapPy to reproduce” anything, including the BLINK tabulation. And if you feel there are hundreds of thousands of tests to be made, then I suggest that you get started making them.

– Marc

]]>I don’t remember the precise statement of the theorem. I believe it’s either in Jaco and Rubinstein’s 0-efficiency paper or in the follow-on “an algorithm to recognise small seifert fibred spaces”. Something like if you have a 0-efficient triangulation of a small Seifert fibred manifold (over S^2 with three singular fibers) then the triangulation is minimal and built from layered solid tori in a prescribed way. That’s the best I can do without a lot of digging, apologies! And of course, you use the algorithm to replace your original triangulation with a 0-efficient one…

]]>Presumably Regina can filter out manifolds with immersed, incompressible tori. How does it recognize elliptic manifolds?

]]>So my statement above was incorrect – you have an enumeration of (closed, oriented, connected) prime 3-manifolds, not hyperbolic ones. Thanks for the clarification.

]]>Frequently there’s very easy heuristics to deal with non-hyperbolic manifolds — for example, SnapPy will often see aspects of non-hyperbolizable manifolds when it gives solutions with negatively oriented tetrahedra.

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