# Determining hyperbolicity of compact orientable 3-manifolds with torus boundary

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 \(T^2\times I.\)