From Hilden-Montesinos theorem, we know that every $3$-Manifold can be constructed as an irregular dihedral 3-fold cover of $S^3$ i.e. there is a represented Knot $K$ such that $ \omega: \pi_1(S^3 - K) \twoheadrightarrow D_3 $ and $M = M(K, \omega)$.
two Questions:
- Is $\pi_1(M) = ker(\omega)$ (ref. to Fox "covering Spaces with singularities" §7)?
- Is $D_3$ the monodromy representation of $\pi_1(S^3 - L)$?
all answers, further questions, references or ideas are welcome :)