How many knots have the same group?

Abstract:

Let K be a knot in ${S^3},G = {\pi _1}({S^3} - K),n =$ number of prime factors of $K,\nu (G) =$ number of topologically different knot-complements with group G and $\kappa (G) =$ number of distinct knot types with group G. Theorem. If K is prime, then $\nu (G) \leqslant 2$. If $n \geqslant 2$, then $\nu (G) = \kappa (G) \leqslant {2^{n - 1}}$. For each $n \geqslant 2$, the bound ${2^{n - 1}}$ is the best possible. For K prime, we still have the conjecture $\nu (G) = \kappa (G) = 1$. If K is a cable-knot, then $\kappa (G) \leqslant 2$.