Consider following program:


1. Generate random 3-manifold embedded in $R^4$.
2. Perform its triangulation.
3. Put it to [Regina](https://en.wikipedia.org/wiki/Regina_(program)) 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](https://arxiv.org/pdf/math/0405250.pdf) 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](https://mathoverflow.net/questions/32787/word-problem-for-fundamental-group-of-submanifolds-of-the-4-sphere) question I found reference to the paper:


*Dranišnikov, A. N.; Repovš, Dušan*, [**Embedding up to homotopy type in Euclidean space**](http://dx.doi.org/10.1017/S0004972700012338), Bull. Aust. Math. Soc. 47, No. 1, 145-148 (1993). [ZBL0796.57011](https://zbmath.org/?q=an:0796.57011).


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,



Random 3-manifolds in $R^4$
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. 


