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Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials

Part of: Topological manifolds

Published online by Cambridge University Press:  12 April 2021

Stefan Friedl ,
Takahiro Kitayama ,
Lukas Lewark ,
Matthias Nagel  and
Mark Powell
Stefan Friedl
Affiliation:
Department of Mathematics, University of Regensburg, Regensburg, Germany e-mail: sfriedl@gmail.com
Takahiro Kitayama
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan e-mail: kitayama@ms.u-tokyo.ac.jp
Lukas Lewark*
Affiliation:
Department of Mathematics, University of Regensburg, Regensburg, Germany e-mail: sfriedl@gmail.com
Matthias Nagel
Affiliation:
Department of Mathematics, ETH Zurich, Zurich, Switzerland e-mail: matthias.nagel@math.ethz.ch
Mark Powell
Affiliation:
Department of Mathematical Sciences, Durham University, Durham, United Kingdom e-mail: mark.a.powell@durham.ac.uk
*
e-mail: lukas@lewark.de
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Abstract

We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine–Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.

Keywords

Ribbon concordanceSeifert formBlanchfield pairingtwisted Alexander polynomialLevine–Tristram signatures

MSC classification

Primary: 57N70: Cobordism and concordance
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 1137 - 1176
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Copyright
© Canadian Mathematical Society 2021

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