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Satellite operators with distinct iterates in smooth concordance


Author: Arunima Ray
Journal: Proc. Amer. Math. Soc. 143 (2015), 5005-5020
MSC (2010): Primary 57M25
DOI: https://doi.org/10.1090/proc/12625
Published electronically: April 20, 2015
MathSciNet review: 3391056
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Abstract: Each pattern $ P$ in a solid torus gives a function $ P:\mathcal {C} \rightarrow \mathcal {C}$ on the smooth knot concordance group, taking any knot $ K$ to its satellite $ P(K)$. We give examples of winding number one patterns $ P$ and a class of knots $ K$, such that the iterated satellites $ P^i(K)$ are distinct in concordance, i.e. if $ i \neq j \geq 0$, $ P^i(K) \neq P^j(K)$. This implies that the operators $ P^i$ give distinct functions on $ \mathcal {C}$, providing further evidence for the (conjectured) fractal nature of $ \mathcal {C}$. Our theorem also allows us to construct several sets of examples, such as infinite families of topologically slice knots that are distinct in smooth concordance, infinite families of 2-component links (with unknotted components and linking number one) which are not smoothly concordant to the positive Hopf link, and infinitely many prime knots which have the same Alexander polynomial as an $ L$-space knot but are not themselves $ L$-space knots.


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Additional Information

Arunima Ray
Affiliation: Department of Mathematics, MS-050, Brandeis University, 415 South St., Waltham, Massachusetts 02453.
Email: aruray@brandeis.edu

DOI: https://doi.org/10.1090/proc/12625
Received by editor(s): April 21, 2014
Received by editor(s) in revised form: September 3, 2014
Published electronically: April 20, 2015
Additional Notes: The author was partially supported by NSF–DMS–1309081 and the Nettie S. Autrey Fellowship (Rice University)
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2015 American Mathematical Society

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