Cappell-Shaneson knot pairs with the same Alexander polynomial
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910094474608640 |
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| author | Endo, Hisaaki Iwaki, Kazunori Pajitnov, Andrei |
| author_facet | Endo, Hisaaki Iwaki, Kazunori Pajitnov, Andrei |
| contents | It is well known that for $m\geq 2$ there are at most two non-equivalent $m$-knots with diffeomorphic exterior. Such pair of knots will be called $\textit{ non-reflexive knot pair}$. A classical problem in topology is to determine all dimensions where such knot pairs exist. In 1976 Cappell and Shaneson gave a method of constructing non-reflexive knot pairs. In the present paper we construct an infinite family of new examples of Cappell-Shaneson knot pairs, and give examples of Cappell-Shaneson knot pairs that have the same Alexander polynomial but are inequivalent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01045 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cappell-Shaneson knot pairs with the same Alexander polynomial Endo, Hisaaki Iwaki, Kazunori Pajitnov, Andrei Geometric Topology Algebraic Topology Number Theory 57K45, 57K10, 11R29 It is well known that for $m\geq 2$ there are at most two non-equivalent $m$-knots with diffeomorphic exterior. Such pair of knots will be called $\textit{ non-reflexive knot pair}$. A classical problem in topology is to determine all dimensions where such knot pairs exist. In 1976 Cappell and Shaneson gave a method of constructing non-reflexive knot pairs. In the present paper we construct an infinite family of new examples of Cappell-Shaneson knot pairs, and give examples of Cappell-Shaneson knot pairs that have the same Alexander polynomial but are inequivalent. |
| title | Cappell-Shaneson knot pairs with the same Alexander polynomial |
| topic | Geometric Topology Algebraic Topology Number Theory 57K45, 57K10, 11R29 |
| url | https://arxiv.org/abs/2604.01045 |