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Transactions of the American Mathematical Society

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Every genus one algebraically slice knot is 1-solvable


Authors: Christopher W. Davis, Taylor Martin, Carolyn Otto and JungHwan Park
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 57M25
DOI: https://doi.org/10.1090/tran/7682
Published electronically: May 30, 2019
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Abstract: Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by $ \mathcal {F}_n$. It has been shown that $ \mathcal {F}_n/\mathcal {F}_{n.5}$ is a very large group for $ n\ge 0$. For a generalization to the setting of links the third author showed that $ \mathcal {F}_{n.5}/\mathcal {F}_{n+1}$ is non-trivial. In this paper we provide evidence for knots $ \mathcal {F}_{0.5}=\mathcal {F}_1$. In particular we prove that every genus 1 algebraically slice knot is 1-solvable.


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Additional Information

Christopher W. Davis
Affiliation: Department of Mathematics, University of Wisconsin–Eau Claire, Eau Claire, Wisconsin 54701
Email: daviscw@uwec.edu

Taylor Martin
Affiliation: Department of Mathematics, Sam Houston State University, Hunstville, Texas 77340
Email: taylor.martin@shsu.edu

Carolyn Otto
Affiliation: Department of Mathematics, University of Wisconsin–Eau Claire, Eau Claire, Wisconsin 54701
Email: ottoa@uwec.edu

JungHwan Park
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: junghwan.park@math.gatech.edu

DOI: https://doi.org/10.1090/tran/7682
Received by editor(s): March 13, 2017
Received by editor(s) in revised form: February 12, 2018
Published electronically: May 30, 2019
Additional Notes: The fourth author was partially supported by the National Science Foundation grant DMS-1309081.
Article copyright: © Copyright 2019 American Mathematical Society