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.
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.
In this question I found reference to the paper:
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.