More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908434684706816 |
|---|---|
| author | Kuhrman, Judson |
| author_facet | Kuhrman, Judson |
| contents | We extend the infinite family of exotic embeddings $\mathbb{RP}^2 \hookrightarrow S^4$ constructed by Miyazawa to a strictly larger family of exotic embeddings, by showing that in place of the pretzel knot $P(-2, 3, 7)$, an infinite family of knots may be used as input to the construction. To this end, we prove that for any Montesinos knot of the form $K(2,3,|6s+1|)$, the branched double cover of the corresponding roll-spun knot is a homotopy sphere. This in turn produces a larger family of homotopy spheres and homotopy $\mathbb{CP}^2$s with potentially interesting involutions. We also observe that Miyazawa's homotopy sphere can be obtained from $S^4$ by a Gluck twist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03798 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres Kuhrman, Judson Geometric Topology 57K45 (Primary) 57K10, 57K40 (Secondary) We extend the infinite family of exotic embeddings $\mathbb{RP}^2 \hookrightarrow S^4$ constructed by Miyazawa to a strictly larger family of exotic embeddings, by showing that in place of the pretzel knot $P(-2, 3, 7)$, an infinite family of knots may be used as input to the construction. To this end, we prove that for any Montesinos knot of the form $K(2,3,|6s+1|)$, the branched double cover of the corresponding roll-spun knot is a homotopy sphere. This in turn produces a larger family of homotopy spheres and homotopy $\mathbb{CP}^2$s with potentially interesting involutions. We also observe that Miyazawa's homotopy sphere can be obtained from $S^4$ by a Gluck twist. |
| title | More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres |
| topic | Geometric Topology 57K45 (Primary) 57K10, 57K40 (Secondary) |
| url | https://arxiv.org/abs/2507.03798 |