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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Linear independence of cables in the knot concordance group


Authors: Christopher W. Davis, JungHwan Park and Arunima Ray
Journal: Trans. Amer. Math. Soc.
MSC (2020): Primary 57K10
DOI: https://doi.org/10.1090/tran/8336
Published electronically: February 11, 2021
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Abstract: We produce infinite families of knots $ \{K^i\}_{i\ge 1}$ for which the set of cables $ \{K^i_{p,1}\}_{i,p\ge 1}$ is linearly independent in the knot concordance group, $ \mathcal {C}$. We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations of  $ \mathcal {C}$, denoted by $ \{\mathcal {F}_n\}$ and $ \{\mathcal {B}_n\}$ respectively. As a consequence, this result cannot be reached by any combination of algebraic concordance invariants, Casson-Gordon invariants, and Heegaard-Floer invariants such as $ \tau $, $ \varepsilon $, and $ \Upsilon $. We give two applications of this result. First, for any $ n\ge 0$, there exists an infinite family $ \{K^i\}_{i\geq 1}$ such that for each fixed $ i$, $ \{K^i_{2^j,1}\}_{j\geq 0}$ is a basis for an infinite rank summand of $ \mathcal {F}_n$ and $ \{K^i_{p,1}\}_{i, p\geq 1}$ is linearly independent in $ \mathcal {F}_{n}/\mathcal {F}_{n.5}$. Second, for any $ n\ge 1$, we give filtered counterexamples to Kauffman's conjecture on slice knots by constructing smoothly slice knots with genus one Seifert surfaces where one derivative curve has nontrivial Arf invariant and the other is nontrivial in both $ \mathcal {F}_n/\mathcal {F}_{n.5}$ and $ \mathcal {B}_{n-1}/\mathcal {B}_{n+1}$. We also give examples of smoothly slice knots with genus one Seifert surfaces such that one derivative has nontrivial Arf invariant and the other is topologically slice but not smoothly slice.


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Additional Information

Christopher W. Davis
Affiliation: Department of Mathematics, University of Wisconsin–Eau Claire, 105 Garfield Avenue P.O. Box 4004, Eau Claire, Wisconsin 54702
Email: daviscw@uwec.edu

JungHwan Park
Affiliation: Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon, 34141, South Korea
Email: jungpark0817@gmail.com

Arunima Ray
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Email: aruray@gmail.com

DOI: https://doi.org/10.1090/tran/8336
Received by editor(s): May 6, 2019
Received by editor(s) in revised form: November 10, 2020
Published electronically: February 11, 2021
Article copyright: © Copyright 2021 American Mathematical Society