Let be a
–manifold
whose boundary consists of tori. The computer program
SnapPea, created by Jeff Weeks, can approximate whether or not
is a complete
hyperbolic manifold. However, until now, there has been no way to determine from this
approximation if
is truly hyperbolic and complete. This paper provides a method for
proving that a manifold has a complete hyperbolic structure based on the
approximations of Snap, a program that includes the functionality of
SnapPea plus other features. The approximation is done by triangulating
,
identifying consistency and completeness equations as described by Neumann and
Zagier [Topology 24 (1985) 307–332] and Benedetti and Petronio [Lectures on
hyperbolic geometry, Universitext, Springer, Berlin (1992)] with respect to this
triangulation, and then, according to Weeks ["Handbook of Knot Theory", Elsevier,
Amsterdam (2005) 461–480], trying to solve the system of equations using Newton’s
Method. This produces an approximate, not actual solution. The method here uses
the Kantorovich Theorem to prove that an actual solution exists, thereby assuring
that the manifold has a complete hyperbolic structure. Using this, we can definitively
prove that every manifold in the SnapPea cusped census has a complete hyperbolic
structure.
