Abstract
A collection \( \Delta \) of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number \( \langle \alpha , \beta \rangle \) is equal to k in absolute value for every \( \alpha , \beta \in \Delta \). Generalizing a theorem of Malestein et al. (Geom Dedicata 168(1):221–233, 2014. doi:10.1007/s10711-012-9827-9) we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is \(2g+1\) when \(g\ge 3\) or k is odd, and 2g otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as \(g^2\).
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Acknowledgements
The authors thank Tarik Aougab, Moira Chas, and Bill Goldman for their enthusiasm and support, and the anonymous referee for their careful reading. We are grateful as well to Josh Greene for his help with an alternative proof of part of our main theorem (see Proposition 9). This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Summer@ICERM 2018: Low Dimensional Topology and Geometry program.
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Daly, C., Gaster, J., Lahn, M. et al. Algebraic k-systems of curves. Geom Dedicata 209, 125–134 (2020). https://doi.org/10.1007/s10711-020-00526-6
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DOI: https://doi.org/10.1007/s10711-020-00526-6