You might find this thread interesting: How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms.

I believe it's expected there should be a polynomial time algorithm to go from a triangulation to a surgery diagram or Heegaard splitting -- in particular see the Thurston and Costantino reference in the above thread. I've been hoping to eventually flesh that out and implement it in Regina but I haven't had the time yet.

For triangulating 3-manifolds given by Heegaard splittings, this was done by Schleimer. See his webpage: http://www.warwick.ac.uk/~masgar/Maths/twister.html there is C-code available, as well. I believe he does something quite comparable to what you describe, using layered triangulations.

You might find this thread interesting: How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms.

I believe it's expected there should be a polynomial time algorithm to go from a triangulation to a surgery diagram or Heegaard splitting -- in particular see the Thurston and Costantino reference in the above thread. I've been hoping to eventually flesh that out and implement it in Regina but I haven't had the time yet.

For triangulating 3-manifolds given by Heegaard splittings, this was done by Schleimer. See his webpage: http://www.warwick.ac.uk/~masgar/Maths/twister.html there is C-code available, as well. I believe he does something quite comparable to what you describe, using layered triangulations.

You might find this thread interesting: How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms.

I believe it's expected there should be a polynomial time algorithm to go from a triangulation to a surgery diagram or Heegaard splitting -- in particular see the Thurston and Costantino reference in the above thread. I've been hoping to eventually flesh that out and implement it in Regina but I haven't had the time yet.

You might find this thread interesting: How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms.

I believe it's expected there should be a polynomial time algorithm to go from a triangulation to a surgery diagram or Heegaard splitting -- in particular see the Thurston and Costantino reference in the above thread. I've been hoping to eventually flesh that out and implement it in Regina but I haven't had the time yet.

For triangulating 3-manifolds given by Heegaard splittings, this was done by Schleimer. See his webpage: http://www.warwick.ac.uk/~masgar/Maths/twister.html there is C-code available, as well. I believe he does something quite comparable to what you describe, using layered triangulations.