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arXiv · 2407.09881

On the genera of symmetric unions of knots

Michel Boileau·Teruaki Kitano·Yuta Nozaki
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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.

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Michel Boileau, Teruaki Kitano, Yuta Nozaki. 2025-06-08. On the genera of symmetric unions of knots. https://doi.org/10.4153/s0008414x25101740

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