  Neumann and Reid conjecture that there are exactly three knot complements
which admit hidden symmetries. This paper establishes several results that
provide evidence for the conjecture. Our main technical tools provide
obstructions to having infinitely many fillings of a cusped manifold produce
knot complements admitting hidden symmetries. Applying these tools, we show for
any two-bridge link complement, at most finitely many fillings of one cusp can
be covered by knot complements admitting hidden symmetries. We also show that
the figure-eight knot complement is the unique knot complement with volume less
than $6v_0 \approx 6.0896496$ that admits hidden symmetries. We then conclude
with two independent proofs that among hyperbolic knot complements only the
figure-eight knot complement can admit hidden symmetries and cover a filling of
the two-bridge link complement $\mathbb{S}^3\setminus 6^2_2$. Each of these
proofs shows that the technical tools established earlier can be made
effective.
