TITLE: Poincare conjecture and the graph of triangulations
QUESTION [6 upvotes]: This was an update to this question, but I decided to make it a separate question. The definition  of the graph of triangulations can be found in the previous question.  
 Question.  I was told a few years ago that some computational complexity problem for the triangulation graph (perhaps in dimension 3?) is related to the Poincare conjecture. Unfortunately I forgot what was the problem. Does anybody know?

REPLY [2 votes]: Perhaps you are thinking of the recognition problem for the three-sphere?  There are theorems of Aleksandar Mijatovic (Simplifying triangulations of $S^3$) and Simon King (How to make a triangulation of $S^3$ polytopal) that say: If $T$ is a triangulation of $S^3$ with at most $n$ tetrahedra then you need at most $f(n)$ "operations" to transform $T$ into a "standard" triangulation.   The function $f$, the operations, and the definition of standard vary between the two papers.  
I believe that the result is a very simple but very slow (doubly exponential) algorithm to recognize the three-sphere.  The proofs, at the very end of the day, rely on normal surface theory and the Rubinstein/Thompson algorithm.