Dehn filling and the knot group II: Ubiquity of persistent elements
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910097040474112 |
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| author | Ito, Tetsuya Motegi, Kimihiko Teragaito, Masakazu |
| author_facet | Ito, Tetsuya Motegi, Kimihiko Teragaito, Masakazu |
| contents | Let $K$ be a nontrivial knot in $S^3$. We say that an element of the knot group $G(K)$ is \textit{persistent} if it remains nontrivial under all nontrivial Dehn fillings. Such elements exist for every nontrivial knot. Indeed, Property P is equivalent to the statement that the meridian of $K$ is a persistent element, and this represents the first instance of such elements. Building on the solution to the Property P conjecture due to Kronheimer and Mrowka, we show that every nontrivial knot group admits infinitely many persistent elements with pairwise disjoint automorphic orbits, none of which contains a power of the meridian. We then develop this further to show that for a broad class of hyperbolic knots - namely those admitting no surgery whose resulting manifold has torsion in its fundamental group - persistent elements are not rare curiosities, but rather structurally pervasive in $G(K)$. This is reflected in the following two properties:
(i) Every subgroup of $G(K)$ that is not contained in the normal closure of a peripheral element contains persistent elements.
(ii) Persistent elements exist outside every proper subgroup of $G(K)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01697 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dehn filling and the knot group II: Ubiquity of persistent elements Ito, Tetsuya Motegi, Kimihiko Teragaito, Masakazu Geometric Topology Group Theory 57M05, 57K10, 57K30, 57M07, 20F65 Let $K$ be a nontrivial knot in $S^3$. We say that an element of the knot group $G(K)$ is \textit{persistent} if it remains nontrivial under all nontrivial Dehn fillings. Such elements exist for every nontrivial knot. Indeed, Property P is equivalent to the statement that the meridian of $K$ is a persistent element, and this represents the first instance of such elements. Building on the solution to the Property P conjecture due to Kronheimer and Mrowka, we show that every nontrivial knot group admits infinitely many persistent elements with pairwise disjoint automorphic orbits, none of which contains a power of the meridian. We then develop this further to show that for a broad class of hyperbolic knots - namely those admitting no surgery whose resulting manifold has torsion in its fundamental group - persistent elements are not rare curiosities, but rather structurally pervasive in $G(K)$. This is reflected in the following two properties: (i) Every subgroup of $G(K)$ that is not contained in the normal closure of a peripheral element contains persistent elements. (ii) Persistent elements exist outside every proper subgroup of $G(K)$. |
| title | Dehn filling and the knot group II: Ubiquity of persistent elements |
| topic | Geometric Topology Group Theory 57M05, 57K10, 57K30, 57M07, 20F65 |
| url | https://arxiv.org/abs/2604.01697 |