## Primary decomposition of knot concordance and von Neumann rho-invariants

HTML articles powered by AMS MathViewer

- by Min Hoon Kim, Se-Goo Kim and Taehee Kim PDF
- Proc. Amer. Math. Soc.
**149**(2021), 439-447 Request permission

## 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 $K$.## References

- Jeff Cheeger and Mikhael Gromov,
*Bounds on the von Neumann dimension of $L^2$-cohomology and the Gauss-Bonnet theorem for open manifolds*, J. Differential Geom.**21**(1985), no. 1, 1–34. MR**806699** - Jae Choon Cha,
*Amenable $L^2$-theoretic methods and knot concordance*, Int. Math. Res. Not. IMRN**17**(2014), 4768–4803. MR**3257550**, DOI 10.1093/imrn/rnt092 - Jae Choon Cha,
*Primary decomposition in the concordance group of topologically slice knots*, arXiv:1910.14629, 2019. - Jae Choon Cha and Kent E. Orr,
*$L^2$-signatures, homology localization, and amenable groups*, Comm. Pure Appl. Math.**65**(2012), no. 6, 790–832. MR**2903800**, DOI 10.1002/cpa.21393 - Tim D. Cochran, Shelly Harvey, and Constance Leidy,
*Primary decomposition and the fractal nature of knot concordance*, Math. Ann.**351**(2011), no. 2, 443–508. MR**2836668**, DOI 10.1007/s00208-010-0604-5 - Tim D. Cochran, Shelly Harvey, and Constance Leidy,
*2-torsion in the $n$-solvable filtration of the knot concordance group*, Proc. Lond. Math. Soc. (3)**102**(2011), no. 2, 257–290. MR**2769115**, DOI 10.1112/plms/pdq020 - Tim D. Cochran, Kent E. Orr, and Peter Teichner,
*Knot concordance, Whitney towers and $L^2$-signatures*, Ann. of Math. (2)**157**(2003), no. 2, 433–519. MR**1973052**, DOI 10.4007/annals.2003.157.433 - Tim D. Cochran, Kent E. Orr, and Peter Teichner,
*Structure in the classical knot concordance group*, Comment. Math. Helv.**79**(2004), no. 1, 105–123. MR**2031301**, DOI 10.1007/s00014-001-0793-6 - Tim D. Cochran and Peter Teichner,
*Knot concordance and von Neumann $\rho$-invariants*, Duke Math. J.**137**(2007), no. 2, 337–379. MR**2309149**, DOI 10.1215/S0012-7094-07-13723-2 - Michael H. Freedman and Frank Quinn,
*Topology of 4-manifolds*, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR**1201584** - Michael H. Freedman,
*A surgery sequence in dimension four; the relations with knot concordance*, Invent. Math.**68**(1982), no. 2, 195–226. MR**666159**, DOI 10.1007/BF01394055 - Hye Jin Jang,
*Two-torsion in the grope and solvable filtrations of knots*, Internat. J. Math.**28**(2017), no. 4, 1750023, 34. MR**3639041**, DOI 10.1142/S0129167X17500239 - Se-Goo Kim,
*Polynomial splittings of Casson-Gordon invariants*, Math. Proc. Cambridge Philos. Soc.**138**(2005), no. 1, 59–78. MR**2127228**, DOI 10.1017/S0305004104008023 - Se-Goo Kim and Taehee Kim,
*Polynomial splittings of metabelian von Neumann rho-invariants of knots*, Proc. Amer. Math. Soc.**136**(2008), no. 11, 4079–4087. MR**2425750**, DOI 10.1090/S0002-9939-08-09372-6 - Se-Goo Kim and Taehee Kim,
*Splittings of von Neumann rho-invariants of knots*, J. Lond. Math. Soc. (2)**89**(2014), no. 3, 797–816. MR**3217650**, DOI 10.1112/jlms/jdu008 - J. Levine,
*Invariants of knot cobordism*, Invent. Math.**8**(1969), 98–110; addendum, ibid. 8 (1969), 355. MR**253348**, DOI 10.1007/BF01404613 - J. Levine,
*Knot cobordism groups in codimension two*, Comment. Math. Helv.**44**(1969), 229–244. MR**246314**, DOI 10.1007/BF02564525

## 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
- © 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