Verified Computations for Hyperbolic 3-Manifolds

Exceptional Dehn surgery on the minimally twisted five-chain link

“…Task (1) was fulfilled via find exceptional fillings.py, a python script written by the author already used in [35] and publicly available [33] to be performed on any cusped hyperbolic three-manifold. The computer-assisted proof is rigorous thanks to the hikmot libraries [23].…”

Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3–manifolds

“…We give the details for N the complement of the knot 10 153 ; the calculations for 10 152 and 10 154 follow along the same lines. The package HIKMOT [17] certifies that the interior of N admits a finite volume hyperbolic metric. Now SnapPy [10] gives the following presentation for the fundamental group of N :…”

The cusped hyperbolic census is complete

“…By combining Theorem 1.1 with the "no false positives" results of Moser [29] and Hoffman et al [22], and by running the latter authors' software HIKMOT over the non-orientable 8-tetrahedron census (which neither paper [22,29] examines), we can finally show that the SnapPea census meets its original aim: Corollary 1.2. The Callahan-Hildebrand-Thistlethwaite-Weeks census tables exactly represent all cusped finite-volume hyperbolic 3-manifolds that can be constructed from n ≤ 8 ideal tetrahedra, with no intruders (false positives) and no omissions (false negatives).…”

“…This possibility can now be eliminated using the techniques of Moser [29] and Hoffman et al [22], who use numerical methods to show that SnapPea's approximate geometric structure is indeed an approximation to an exact geometric structure. In particular, Moser has shown that the n ≤ 7-tetrahedron census contains no false positives, and Hoffman et al have shown that the orientable n ≤ 8-tetrahedron census contains no false positives.…”

“…Most computations from this paper are performed using the software package Regina [5,13]; some computations also use HIKMOT [22], Magma [3] and SnapPy [18], and these are noted where they occur.…”