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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- Min Hoon Kim
- Affiliation: Department of Mathematics, Chonnam National University, Gwangju 61186, Republic of Korea
- MR Author ID: 1067137
- Email: firstname.lastname@example.org
- 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: email@example.com
- Taehee Kim
- Affiliation: Department of Mathematics, Konkuk University, Seoul 05029, Republic of Korea
- MR Author ID: 743933
- Email: firstname.lastname@example.org
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 439-447
- MSC (2020): Primary 57K10, 57K31, 57K40, 57N70
- DOI: https://doi.org/10.1090/proc/15282
- MathSciNet review: 4172618