For canonical ideal triangulation of a hyperbolic knot, the symmetries of the knot are the same as the symmetries of that triangulation. This is how SnapPy computes the knot symmetry group.
Is there some relation between the symmetries of a hyperbolic knot and the symmetries of a generic ideal triangulation (potentially with different numbers of tetrahedrons than the canonical one)? In particular, I wonder if the symmetry of the triangulation always leads to a symmetry of knot.
Meanwhile, for an ideal triangulation, one can write down gluing equations for the internal edges. What's the relationship between the symmetries of gluing equations and the symmetries of knots?
