Random 3-manifolds in $R^4$ Consider following program:


*

*Generate random 3-manifold embedded in $R^4$.

*Perform its triangulation.

*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,
 A: 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. 
