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Primary decomposition of knot concordance and von Neumann rho-invariants


Authors: Min Hoon Kim, Se-Goo Kim and Taehee Kim
Journal: Proc. Amer. Math. Soc. 149 (2021), 439-447
MSC (2020): Primary 57K10, 57K31, 57K40, 57N70
DOI: https://doi.org/10.1090/proc/15282
Published electronically: October 20, 2020
MathSciNet review: 4172618
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Abstract: We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher order von Neumann $\rho$-invariants by Cochran, Orr, and Teichner. We show that for a non-negative integer $n$, if the connected sum of two $n$-solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann $\rho$-invariants of order $n$. This gives positive evidence for the conjecture that nonslice knots with coprime Alexander polynomials are not concordant. As an application, we show that if $K$ is one of Cochran–Orr–Teichner’s knots which are the first examples of nonslice knots with vanishing Casson–Gordon invariants, then $K$ is not concordant to any knot with Alexander polynomial coprime to that of $K$.


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Additional Information

Min Hoon Kim
Affiliation: Department of Mathematics, Chonnam National University, Gwangju 61186, Republic of Korea
MR Author ID: 1067137
Email: minhoonkim@jnu.ac.kr

Se-Goo Kim
Affiliation: Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 02447, Republic of Korea
MR Author ID: 610250
ORCID: 0000-0002-8874-9408
Email: sgkim@khu.ac.kr

Taehee Kim
Affiliation: Department of Mathematics, Konkuk University, Seoul 05029, Republic of Korea
MR Author ID: 743933
Email: tkim@konkuk.ac.kr

Received by editor(s): November 29, 2019
Received by editor(s) in revised form: May 15, 2020
Published electronically: October 20, 2020
Additional Notes: The first named author was partly supported by NRF grant 2019R1A3B2067839.
The second named author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07047860).
The last named author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (no.2018R1D1A1B07048361).
The third author is the corresponding author.
Communicated by: David Futer
Article copyright: © Copyright 2020 American Mathematical Society