Embedding infinite cyclic covers of knot spaces into 3-space
- Creators
- Jiang, Boju
- Ni, Yi
- Wang, Shicheng
- Zhou, Qing
Abstract
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.
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.Attached Files
Submitted - 0505206.pdf
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Additional details
- Eprint ID
- 56831
- DOI
- 10.1016/j.top.2006.01.005
- Resolver ID
- CaltechAUTHORS:20150421-115920260
- MOSTC
- MOEC
- Princeton University Centennial Fellowship
- Created
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2015-04-21Created from EPrint's datestamp field
- Updated
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2021-11-10Created from EPrint's last_modified field