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Consider following program:

  1. Generate random 3-manifold embedded in $R^4$.
  2. Perform its triangulation.
  3. Put it to Regina and calculate what manifold it is.

Assuming that we have good algorithm for random submanifolds in point 1. then we can conclude which 3-manifolds of of complexity 5,6,7,8 etc are embeddable in $R^4$. For example from this paper I can see that there are 175 3-manifolds of complexity 7. Those which were not obtained in this process we can assume are not embeddable in $R^4$ with some probability.

Possible choices for algorithm in point 1 are:

a) zero of four variables polynomial;

b) random embedded 1-surgery;

c) gluing cubes;

d) drilling small hole cubes in big cube;

e) boundary of regular neighborhood of 2-complex in $R^4$ (added 2018-08-23)

The questions are:

A. What are achievements in finding good polynomial of four variables hoping to obtain interesting 3-manifold as its zero ?

B. What could be the algorithm for finding random loop in $M$ embedded in $R^4$ to perform embedded surgery ?

C. Is Regina accepting command line execution with some input in TXT file containing triangulation and producing result (or LOG) in other TXT file ?

Related questions are:

D. What could be other ideas for producing random 3-submanifolds of $R^4$ ?

E. How could we generate random slice knots and what manifolds we obtain by repeating 1-surgery on slice knots ?

F. Is it known which 2-dimensional CW-complexes are embeddable in $R^4$ ? Such CW-complex can be seen as few words in set of generators which are forming bouquet of circles. I am hoping all 3-manifolds embeddable in 4-space are boundaries of regular neighborhoods of some 2-complex.

EDIT 2018-07-30

Regarding last question. I have been able to find embedding of 2-complex with one word in 4-space. So I thought to use this as starting point. Assuming that this 2-complex is defined as 2-skeleton in $\mathbb R^4$. Related question is

F2. Is it known algorithm for finding regular neighborhood of 2-skeleton in $R^4$ ? If I have it then I find its triangulated boundary as 3-manifold I want. The 3-simplex belongs to boundary when it belongs to only one 4-simplex.

EDIT 2018-08-23

In this question I found reference to the paper:

Dranišnikov, A. N.; Repovš, Dušan, Embedding up to homotopy type in Euclidean space, Bull. Aust. Math. Soc. 47, No. 1, 145-148 (1993). ZBL0796.57011.

In this paper there is construction of embedding of any 2-complex in $R^4$ up to homotopy type. It is described as simpler proof of Stallings theorem from 1965. Therefore I am planning to convert that construction to simplicial complex in $R^4$. Next construct its regular neighborhood, its boundary will be 3-manifold which I would like to recognize using Regina or other software.

What is not clear for me is why embedding of 2-complex in $R^4$ listed as open issue number 5.3 on Kirby's open problem list.

Regards,

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    $\begingroup$ What the hell is Regina? :D $\endgroup$ – Qfwfq Jul 16 '18 at 16:54
  • $\begingroup$ (I didn't downvote, for the record..) $\endgroup$ – Qfwfq Jul 16 '18 at 17:12
  • $\begingroup$ I added link to wikipedia page for Regina software. It calculates fundamental group, homology groups and determine type of 3-manifold based on its triangulation. $\endgroup$ – Marek Mitros Jul 16 '18 at 17:20
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    $\begingroup$ Have you looked at Budney-Burton arxiv.org/abs/0810.2346v5? It investigates subcomplexes of $R^4$ via Regina and so seems related to some of what you ask for. As to question F, I think this is not known. There are some comments in Kirby's problem list (math.berkeley.edu/~kirby/problems.ps.gz) on the subject (see problem 5.3, attributed to R. Fenn). Also, in problem F it's more correct to refer to a regular neighborhood rather than tubular neighborhood. $\endgroup$ – Danny Ruberman Jul 16 '18 at 19:20
  • $\begingroup$ Regarding C, see Regina's python interface regina-normal.github.io/docs/python.html . There is a command-line interface regina-python for macOS and linux, but not windows, apparently. $\endgroup$ – j.c. Jul 17 '18 at 7:26
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Regarding A: As far as I know, there's only some special cases and no big familiy of interesting examples known.

Regarding C: Yes, Regina has a fairly good Python interface. We don't have every feature of the C++ library implemented in Python, but quite a bit is.

Regarding D: Perhaps the most sensible way to define a "random submanifold of $R^4$" would be to start with the standard triangulation of $S^4$, and sequentially do barycentric subdivision. A "random" submanifold could be a vertex-normal solution to the normal 3-dimensional submanifold equations. This is something I do with Regina systematically and it's one of the more fruitful ways of generating 3-dimensional submanifolds of $S^4$. Unfortunately, you generate the simplest 3-manifolds the most often, much like with random knot algorithms. It's unclear to me how to make this more effective, other than throwing enormous computational power at the problem.

Regarding E: I don't think people have any great algorithms for producing random slice knots. You either have to restrict to fairly specific knot families or live with cripplingly slow algorithms.

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  • $\begingroup$ Thank you for the answer ! I accepted it. I am reading Regina Handbook to understand what is "normal vertex hypersurface" in 4-manifold. $\endgroup$ – Marek Mitros Jul 30 '18 at 8:24
  • $\begingroup$ I peform following steps in Regina: - New Topology Data, New 4-Triangulation (4-sphere minimal); created 2 pentachorons. Select barycentric subdivision; 240 pentachorons are present. Next click icon New Normal Hypersurface List. The dialog Working is shown with description "Enumerating vertex hypersurfaces (double description method). The progress reach 2% and does not move. I click "Cancel" after half an hour. I am using Windows version of Regina 5.1 $\endgroup$ – Marek Mitros Jul 30 '18 at 10:41
  • $\begingroup$ Right. Normal 3-manifold enumeration in a 240 pentachoron sphere will be horrifically slow. To avoid these kinds of issues what I tend to do is more of a random walk on the Pachner graph -- i.e. I do various Pachner moves to complicate the triangulation. If you keep it under 24 or so pentachora, enumeration should be fine on "normal" home computers. $\endgroup$ – Ryan Budney Jul 30 '18 at 22:48
  • $\begingroup$ Thanks for explanation. Now I am more willing to use GAP package "simpcomp" for work with simplicial complexes. My program is now: 1) take 2-complex; 2) calculate regular neighborhood in $R^4$ (in this point not certain whether 2-complex can be embedded there); 3) calculate homology of simplicial complex obtained this way; 4) look around for other 3-manifold recognition methods. $\endgroup$ – Marek Mitros Jul 31 '18 at 5:54
  • $\begingroup$ I noticed "drill edge" function in Regina, so I am now considering Regina. How can I perform 1-surgery on 3-manifold using path in 1-skeleton ? First I drill all edges on this path. But next how should I glue torus ? $\endgroup$ – Marek Mitros Aug 3 '18 at 14:03

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