Posted as arXiv:1410.7115.
Published in JoCG 11 (1), 125--136, 2020.
I wrote, proved correct, and implemented as a Python extension of Regina a test for the hyperbolicity of nontrivial generalized link exteriors. I ran this algorithm on 762 three-manifolds in order to resolve in the affirmative a conjecture of Gabai, Meyerhoff, and Milley from 2011.
Apart from the Regina software suite, the main thing that made this feasible to implement was a new simple test for homeomorphism to \(T^2 \times I.\) Here is my implementation in Python:
def isT2xI(T): if not isHomologyT2xI(T): return False X = clone(T) X.intelligentSimplify() simplifyBoundary(X) k = X.boundaryComponent(0) for e in k.faces(1): i = e.index() Xe = clone(X) Xe.closeBook(Xe.face(1,i)) Xe.intelligentSimplify() if not Xe.isSolidTorus(): return False else: return True
It fits on the back of a napkin and requires no boundary patterns. Just close-the-book along three edges and test for homeomorphism to a solid torus. If any isn’t a solid torus, then the manifold wasn’t \(T^2\times I.\) If they all are, then by seminal work of Berge and Gabai, the manifold must be \(T^2\times I.\)