Dehn Filling and the Knot Group I: Realization Property

Abstract Given a non-trivial knot $K$ in $S^{3}$, its exterior $E(K)$ admits infinitely many Dehn fillings parametrized by slopes $r \in \mathbb{Q}$, each of which yields a closed $3$-manifold $K(r)$ and trivializes some elements in the knot group $G(K) = \pi _{1}(E(K))$ via the induced homomorphism from $G(K)$ onto $\pi _{1}(K(r))$. For hyperbolic knots $K$, it is known that every non-trivial element of $G(K)$ remains non-trivial for all but finitely many Dehn fillings. We address the question: Given finitely many slopes $r_{1}, \dots , r_{n}$, does there exist an element in $G(K)$ such that it becomes trivial after $r_{i}$-Dehn fillings for these pre-specified slopes $r_{1}, \dots , r_{n}$, while it remains non-trivial for all other non-trivial Dehn fillings? In this article, we answer this question in the positive for most hyperbolic knots, including those without exceptional surgeries. Furthermore, we show that there are infinitely many such elements up to conjugacy and powers.