Nontrivial Elements in a Knot Group That are Trivialized by Dehn Fillings

Abstract

Abstract Let $K$ be a nontrivial knot in $S^3$ with the exterior $E(K)$, and $\gamma \in G(K) = \pi _1(E(K), *)$ a slope element represented by an essential simple closed curve on $\partial E(K)$ with base point $* \in \partial E(K)$. Since the normal closure $\langle \!\langle \gamma \rangle \!\rangle $ of $\gamma $ in $G(K)$ coincides with that of $\gamma ^{-1}$, and $\gamma $ and $\gamma ^{-1}$ correspond to a slope $r \in \mathbb{Q} \cup \{ \infty \}$, we write $\langle \!\langle r \rangle \!\rangle = \langle \!\langle \gamma \rangle \!\rangle $. The normal closure $\langle \!\langle r \rangle \!\rangle $ describes elements, which are trivialized by $r$-Dehn filling of $E(K)$. In this article, we prove that $\langle \!\langle r_1 \rangle \!\rangle = \langle \!\langle r_2 \rangle \!\rangle $ if and only if $r_1 = r_2$, and for a given finite family of slopes $\mathcal{S} = \{ r_1, \dots , r_n \}$, the intersection $\langle \!\langle r_1 \rangle \!\rangle \cap \cdots \cap \langle \!\langle r_n \rangle \!\rangle $ contains infinitely many elements except when $K$ is a $(p, q)$-torus knot and $pq \in \mathcal{S}$. We also investigate inclusion relation among normal closures of slope elements.