The Plain Sphere Number of a Link

Ryan Blair; Alexandra Kjuchukova; Ella Pfaff

doi:10.2140/pjm.2026.342.217

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arXiv · 2504.10517

The Plain Sphere Number of a Link

Ryan Blair·Alexandra Kjuchukova·Ella Pfaff

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Abstract

Let $L$ be a link in $S^3$. We consider the class of meridional presentations for $\pi_1(S^3\backslash L)$ in which the relations are witnessed by embedded two-spheres which can be represented simultaneously in a fixed diagram of $L$, analogously to decomposition spheres studied by Cromwell, Menasco and others. Wirtinger relations are witnessed by such spheres and the Wirtinger presentation is a special case of the ones we study. We prove that the smallest number of generators of $\pi_1(S^3\backslash L)$ over all such presentations equals the bridge number of $L$.

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Ryan Blair, Alexandra Kjuchukova, Ella Pfaff. 2025-04-10. The Plain Sphere Number of a Link. https://doi.org/10.2140/pjm.2026.342.217

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