overlapping of tetrahedra. From this description of the hyperbolic structure on a link complement, SnapPea can then perform hyperbolic Dehn filling on the cusps to obtain more hyperbolic 3-manifolds. SnapPea does this by taking any given slopes which determine certain Dehn filling equations (also explained in Thurston's notes), and then adjusting the shapes of the ideal tetrahedra to give solutions to these equations and the edge equations. For almost all slopes, this gives an incomplete hyperbolic structure on the link complement, whose completion gives a hyperbolic structure on the Dehn-filled manifold. Its volume is the sum of the volumes of the adjusted tetrahedra. Canonical decomposition SnapPea is usually able to compute the canonical decomposition of a cusped hyperbolic 3-manifold from a given ideal triangulation. If not, then it randomly retriangulates and tries again. This has never been known to fail. The canonical decomposition allows SnapPea to tell two cusped hyperbolic 3-manifolds apart by turning the problem of recognition into a combinatorial question, i.e. checking if the two manifolds have combinatorially equivalent canonical decompositions. SnapPea is also able to check if two closed hyperbolic 3-manifolds are isometric by drilling out short geodesics to create cusped hyperbolic 3-manifolds and then using the canonical decomposition as before. The recognition algorithm allow SnapPea to tell two hyperbolic knots or links apart. Weeks, et al., were also able to compile different censuses of hyperbolic 3-manifolds by using the algorithm to cull lists of duplicates. Additionally, from the canonical decomposition, SnapPea is able to: Compute the Ford domain Compute the symmetry group Computable invariants Censuses SnapPea has several databases of hyperbolic 3-manifolds available for systematic study. Cusped census Closed census See also Regina incorporates aspects of SnapPea. Computational topology References External links SnapPea Jeff Weeks' site SnapPy Culler and Dunfield's extension Orb Damian Heard's extension, allows : hyperbolic manifolds with totally geodesic boundary orbifolds where the orbifold locus contains trivalent vertices 3-manifolds Computational topology Numerical software Free software programmed in C Free mathematics software