The author describes an effective algorithm for finding a counterexample to the Poincaré conjecture. This is a consequence of two algorithms:
(1) The Rêgo-Rourke algorithm which lists all Heegaard diagrams of homotopy 3-spheres.
(2) The Rubinstein-Thompson algorithm which will decide if a given triangulated 3-manifold is homeomorphic to \(S^3\).
A Heegaard diagram can be converted into a triangulation in several algorithmic ways and combining all three algorithms the author obtains the following result:
Main Theorem. There is an algorithm with no input data which lists all nontrivial homotpy 3-spheres. Thus, if the Poincaré conjecture is false, the algorithm will produce a proven counterexample in finite time. (If it is true, it will continue forever with no results.)
The author briefly sketches the two main algorithms (of Rêgo-Rourke and Rubinstein-Thompson) and finishes his paper with some comments on the practicality of the combined algorithm.