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Generalized crossing changes in satellite knots


Author: Cheryl Jaeger Balm
Journal: Proc. Amer. Math. Soc. 143 (2015), 447-458
MSC (2010): Primary 57M27
DOI: https://doi.org/10.1090/S0002-9939-2014-12235-0
Published electronically: August 18, 2014
MathSciNet review: 3272768
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Abstract: We show that if $ K$ is a satellite knot in the 3-sphere $ S^3$ which admits a generalized cosmetic crossing change of order $ q$ with $ \vert q\vert \geq 6$, then $ K$ admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot in $ S^3$ which admits a torus knot as a pattern cannot admit a generalized cosmetic crossing change of order $ q$ with $ \vert q\vert \geq 6$. We also show that if there is any knot in $ S^3$ admitting a generalized cosmetic crossing change of order $ q$ with $ \vert q\vert \geq 6$, then there must be such a knot which is hyperbolic.


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Additional Information

Cheryl Jaeger Balm
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: balmcher@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12235-0
Received by editor(s): October 23, 2012
Received by editor(s) in revised form: March 14, 2013, and March 19, 2013
Published electronically: August 18, 2014
Additional Notes: This research was supported by NSF grant DMS-1105843
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.