Dehn Surgery and Hyperbolic Knot Complements without Hidden Symmetries

Abstract

Abstract Neumann and Reid conjectured that only three hyperbolic knot complements admit hidden symmetries. Here, we provide evidence for the conjecture, giving obstructions for a manifold to have infinitely many fillings that are knot complements with hidden symmetries. Applying these, we show that at most finitely many fillings of any hyperbolic two-bridge link complement can be covered by knot complements with hidden symmetries. We then make our tools effective, showing first that the only knot complement with hidden symmetries and volume less than $6v_0 \approx 6.0896496$ is the complement of the figure-eight. We conclude with two proofs that if a hyperbolic knot’s complement admits hidden symmetries and covers a filling of the complement of the $6_2^2$ link, it is the figure-eight.