Almost-concordance of knots in aspherical 3-manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913998640775168 |
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| author | Stees, Ryan |
| author_facet | Stees, Ryan |
| contents | In this paper, we study topological concordance modulo local knotting, or almost-concordance, of knots in 3-manifolds $M\neq S^3$. A. Levine, Celoria (arXiv:1602.05476v4), and Friedl-Nagel-Orson-Powell (arXiv:1611.09114v2) conjecture that, absent the presence of an embedded dual 2-sphere, any free homotopy class $x$ of knots in $M$ contains infinitely many concordance classes modulo the action of the concordance group of knots in $S^3$ by local knotting. We develop a method for confirming this conjecture for any nontrivial class $x$ in any aspherical $M$ and provide computations that prove the conjecture in a large family of open cases. Our technique employs an extension of Milnor's link invariants to knots and links in non-simply-connected 3-manifolds (arXiv:2310.10918v2). We exhibit a large family of examples where, in a precise sense, we maximize the number of almost-concordance classes distinguished by these invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost-concordance of knots in aspherical 3-manifolds Stees, Ryan Geometric Topology 57K10, 57K30, 57K40 In this paper, we study topological concordance modulo local knotting, or almost-concordance, of knots in 3-manifolds $M\neq S^3$. A. Levine, Celoria (arXiv:1602.05476v4), and Friedl-Nagel-Orson-Powell (arXiv:1611.09114v2) conjecture that, absent the presence of an embedded dual 2-sphere, any free homotopy class $x$ of knots in $M$ contains infinitely many concordance classes modulo the action of the concordance group of knots in $S^3$ by local knotting. We develop a method for confirming this conjecture for any nontrivial class $x$ in any aspherical $M$ and provide computations that prove the conjecture in a large family of open cases. Our technique employs an extension of Milnor's link invariants to knots and links in non-simply-connected 3-manifolds (arXiv:2310.10918v2). We exhibit a large family of examples where, in a precise sense, we maximize the number of almost-concordance classes distinguished by these invariants. |
| title | Almost-concordance of knots in aspherical 3-manifolds |
| topic | Geometric Topology 57K10, 57K30, 57K40 |
| url | https://arxiv.org/abs/2508.14638 |