Nontrivial Elements in a Knot Group That are Trivialized by Dehn Fillings
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 \!\ran...
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Veröffentlicht in: | International mathematics research notices 2021-06, Vol.2021 (11), p.8297-8321 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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. |
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ISSN: | 1073-7928 1687-0247 |
DOI: | 10.1093/imrn/rnz069 |