Osaka Journal of Mathematics

Self-intersections of curves on a surface and Bernoulli numbers

Shinji Fukuhara, Nariya Kawazumi, and Yusuke Kuno

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We study an operation which measures self-intersections of curves on an oriented surface. It turns out that a certain computation on this topological operation is related to the Bernoulli numbers $B_m$, and our study yields a family of explicit formulas for $B_m$. As a special case, this family contains the celebrated formula for $B_m$ due to Kronecker.

Article information

Osaka J. Math., Volume 55, Number 4 (2018), 761-768.

First available in Project Euclid: 10 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B68: Bernoulli and Euler numbers and polynomials 57N05: Topology of $E^2$ , 2-manifolds 57M99: None of the above, but in this section


Fukuhara, Shinji; Kawazumi, Nariya; Kuno, Yusuke. Self-intersections of curves on a surface and Bernoulli numbers. Osaka J. Math. 55 (2018), no. 4, 761--768.

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