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On embedding the infinite cyclic coverings of knot complements into three sphere

Author(s): Zhiqing Yang
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 57M25; Secondary 57M05
Posted: October 28, 2009
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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:

1.
C. McA. Gordon, On embedding infinite cyclic covers in compact 3-manifolds, preprint, Math. GT/0608339, http://front.math.ucdavis.edu/0608.5339.

2.
Boju Jiang, Yi Ni, Shicheng Wang and Qing Zhou, Embedding infinite cyclic covers of knot spaces into 3-space, Topology, vol. 45, issue 4, July 2006, 691-705. MR 2236373 (2007g:57010)

3.
W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Wiley, 1966 (reprinted by Dover, 1976). MR 0422434 (54:10423)

4.
D. Rolfsen, Knots and links, Publish or Perish Inc., Berkeley, 1976, 160-197. MR 0515288 (58:24236)

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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

DOI: 10.1090/S0002-9939-09-10137-5
PII: S 0002-9939(09)10137-5
Keywords: infinite cyclic covering, knot, Seifert surface, knot group
Received by editor(s): July 26, 2008,
Received by editor(s) in revised form: February 26, 2009, July 17, 2009, and July 29, 2009
Posted: 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 of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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