When are two hyperbolic knots equal?

We will show that complements of hyperbolic knots can be uniquely triangulated and it is possible to distingush between these knots by combinatorically comparing their triangulations. This fact follows from the following sequence of theorems: the Gordon-Luecke theorem on knot complements, the Mostow rigidity theorem, and the Epstein-Penner-Weeks theorem on the Euclidean decomposition of cusped 3-manifolds. We will also discuss knot symmetry types and show that the unique triangulation also allows us to compute the knot symmetry group.