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On the genera of symmetric unions of knots

Part of: Low-dimensional topology Differential topology

Published online by Cambridge University Press:  03 November 2025

Michel Boileau ,
Teruaki Kitano Open the ORCID record for Teruaki Kitano [Opens in a new window]  and
Yuta Nozaki Open the ORCID record for Yuta Nozaki [Opens in a new window]
Michel Boileau
Affiliation:
Aix-Marseille University , CNRS, Centrale Marseille, I2M, UMR 7373, France e-mail: michel.boileau@univ-amu.fr
Teruaki Kitano
Affiliation:
Department of Information Systems Science, Faculty of Science and Engineering, Soka University , Japan e-mail: kitano@soka.ac.jp
Yuta Nozaki*
Affiliation:
Faculty of Environment and Information Sciences, Yokohama National University , Yokohama, Japan WPI-SKCM2, Hiroshima University , Japan
*
e-mail: nozaki-yuta-vn@ynu.ac.jp
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Abstract

In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.

Keywords

Symmetric unionknot groupgenustwisted Alexander polynomial

MSC classification

Secondary: 57M05: Fundamental group, presentations, free differential calculus 57R18: Topology and geometry of orbifolds

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 26
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This study was supported in part by JSPS KAKENHI Grant Numbers JP19K03505, JP20K14317, and JP23K12974. The first and second authors were supported by the Soka University International Collaborative Research Grant.

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