On embedding the infinite cyclic coverings of knot complements into three sphere
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Abstract:
We construct a class of knots with the CI${}^*$ property, that is, $\pi _1(M(n)\mid \partial M(n))\neq \{e\}$ for some $n>0$. It follows that the infinite cyclic covering of such a knot cannot be embedded in any compact 3-manifold.References
- C. McA. Gordon, On embedding infinite cyclic covers in compact 3-manifolds, preprint, Math. GT/0608339, http://front.math.ucdavis.edu/0608.5339.
- Boju Jiang, Yi Ni, Shicheng Wang, and Qing Zhou, Embedding infinite cyclic covers of knot spaces into 3-space, Topology 45 (2006), no. 4, 691–705. MR 2236373, DOI 10.1016/j.top.2006.01.005
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. MR 0422434
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Additional Information
- Zhiqing Yang
- Affiliation: School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, People’s Republic of China
- Email: yangzhq@dlut.edu.cn
- Received by editor(s): July 26, 2008
- Received by editor(s) in revised form: February 26, 2009, July 17, 2009, and July 29, 2009
- Published electronically: October 28, 2009
- Additional Notes: The author is supported by a grant (No. 100771023) of NSFC and a grant from Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (20070141035).
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1153-1157
- MSC (2010): Primary 57M25; Secondary 57M05
- DOI: https://doi.org/10.1090/S0002-9939-09-10137-5
- MathSciNet review: 2566580