## Higher-order Alexander invariants and filtrations of the knot concordance group

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- by Tim D. Cochran and Taehee Kim PDF
- Trans. Amer. Math. Soc.
**360**(2008), 1407-1441 Request permission

## Abstract:

We establish certain “nontriviality” results for several filtrations of the smooth and topological knot concordance groups. First, as regards the*n-solvable filtration*of the topological knot concordance group, $\mathcal {C}$, defined by K. Orr, P. Teichner and the first author: \[ 0\subset \cdots \subset \mathcal {F}_{(n.5)}\subset \mathcal {F}_{(n)}\subset \cdots \subset \mathcal {F}_{(1.5)}\subset \mathcal {F}_{(1.0)}\subset \mathcal {F}_{(0.5)} \subset \mathcal {F}_{(0)}\subset \mathcal {C},\] we refine the recent nontriviality results of Cochran and Teichner by including information on the Alexander modules. These results also extend those of C. Livingston and the second author. We exhibit similar structure in the closely related

*symmetric Grope filtration*of $\mathcal {C}$. We also show that the Grope filtration of the

*smooth*concordance group is nontrivial using examples that cannot be distinguished by the Ozsváth-Szabó $\tau$-invariant nor by J. Rasmussen’s $s$-invariant. Our broader contribution is to establish, in “the relative case”, the key homological results whose analogues Cochran-Orr-Teichner established in “the absolute case”. We say two knots $K_0$ and $K_1$ are

*concordant modulo $n$-solvability*if $K_0\#(-K_1)\in \mathcal {F}_{(n)}$. Our main result is that, for any knot $K$ whose classical Alexander polynomial has degree greater than 2, and for any positive integer $n$, there exist infinitely many knots $K_i$ that are concordant to $K$ modulo $n$-solvability, but are all distinct modulo $n.5$-solvability. Moreover, the $K_i$ and $K$ share the same classical Seifert matrix and Alexander module as well as sharing the same

*higher-order*

*Alexander modules*and Seifert presentations up to order $n-1$.

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## Additional Information

**Tim D. Cochran**- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005-1892
- Email: cochran@math.rice.edu
**Taehee Kim**- Affiliation: Department of Mathematics, Konkuk University, Seoul 143-701, Korea
- MR Author ID: 743933
- Email: tkim@konkuk.ac.kr
- Received by editor(s): January 26, 2005
- Received by editor(s) in revised form: October 21, 2005
- Published electronically: October 5, 2007
- Additional Notes: The first author was partially supported by the National Science Foundation
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**360**(2008), 1407-1441 - MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9947-07-04177-3
- MathSciNet review: 2357701