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Embedding infinite cyclic covers of knot spaces into 3-space

Jiang, Boju and Ni, Yi and Wang, Shicheng and Zhou, Qing (2006) Embedding infinite cyclic covers of knot spaces into 3-space. Topology, 45 (4). pp. 691-705. ISSN 0040-9383. doi:10.1016/

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We say a knot k in the 3-sphere S^3 has Property IE if the infinite cyclic cover of the knot exterior embeds into 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 Δk(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 _(Δk)(t)=1 and 2t^2−5t+2 respectively. Hence among genus 1 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.

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Additional Information:© 2006 Elsevier Ltd. Received 5 December 2004. Dedicated to the memory of Professor Shiing-shen Chern. We are grateful to Dr. Hao Zheng for drawing the pictures, to Professor William Browder for a helpful conversation with the second author, to Professor Robert D. Edwards for bringing the third author to this simply stated intuitive question, and to Professor David Gabai for a comment on alternating knots. All authors are partially supported by a MOSTC grant and a MOEC grant. The second author is partially supported by the Centennial fellowship of Princeton University. Part of the paper is revised from the Master thesis of the second named author submitted to Peking University.
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Princeton University Centennial FellowshipUNSPECIFIED
Subject Keywords:Embedding; Non-fibre knots; Infinite cyclic coverings; Alexander polynomial
Issue or Number:4
Classification Code:MSC: 57M10; 57M25; 57N30
Record Number:CaltechAUTHORS:20150421-115920260
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Official Citation:Boju Jiang, Yi Ni, Shicheng Wang, Qing Zhou, Embedding infinite cyclic covers of knot spaces into 3-space, Topology, Volume 45, Issue 4, July 2006, Pages 691-705, ISSN 0040-9383, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:56831
Deposited By: George Porter
Deposited On:21 Apr 2015 19:41
Last Modified:10 Nov 2021 21:04

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