Embedding infinite cyclic covers of knot spaces into 3-space
| dc.creator | Jiang, Boju | |
| dc.creator | Ni, Yi | |
| dc.creator | Wang, Shicheng | |
| dc.creator | Zhou, Qing | |
| dc.date | 2005-05-11 | |
| dc.date.accessioned | 2026-07-07T05:19:47Z | |
| dc.date.available | 2026-07-07T05:19:47Z | |
| dc.description | We say a knot $k$ in the 3-sphere $\mathbb S^3$ has {\it Property $IE$} if the infinite cyclic cover of the knot exterior embeds into $\mathbb S^3$. Clearly all fibred knots have Property $IE$. There are infinitely many non-fibred knots with Property $IE$ and infinitely many non-fibred knots without property $IE$. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property $IE$, then its Alexander polynomial $Δ_k(t)$ must be either 1 or $2t^2-5t+2$, and we give two infinite families of non-fibred genus 1 knots with Property $IE$ and having $Δ_k(t)=1$ and $2t^2-5t+2$ respectively. Hence among genus one non-fibred knots, no alternating knot has Property $IE$, and there is only one knot with Property $IE$ up to ten crossings. We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold. | |
| dc.description | 24 pages, 9 figures | |
| dc.identifier | https://arxiv.org/abs/math/0505206 | |
| dc.identifier | http://arxiv.org/abs/math/0505206 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75147 | |
| dc.subject | Geometric Topology | |
| dc.subject | Dynamical Systems | |
| dc.subject | 57M10, 57M25, 57N30, 37E99 | |
| dc.title | Embedding infinite cyclic covers of knot spaces into 3-space | |
| dc.type | text |