Abstract: A hyperbolic manifold M admits hidden symmetries if it has a cover M̂ that normally covers a smaller volume quotient than M itself. In the case of hyperbolic knot complements, having hidden symmetries is seemingly quite rare. In fact, there are only three knot complements known to admit hidden symmetries: the figure 8 knot complement and two knot complements which decompose into regular ideal dodecahedra.
All three can be realized by small surgeries on fully augmented links. In contrast, we will show that knots obtained from high parameter surgeries on fully augmented links cannot admit hidden symmetries. Furthermore, we can place strong restrictions on the (standard) symmetries of these knots.
This is joint work with C. Millichap and W. Worden.