Constructing knots with low rational genera

Clayton McDonald; Allison N. Miller

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arXiv · 2511.15900

Constructing knots with low rational genera

Clayton McDonald·Allison N. Miller

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Abstract

We give a flexible construction for knots in the 3-sphere that bound surfaces of unexpectedly low genus in punctured open books on 3-manifolds. We use this construction to give the first examples of knots whose genus differs in different $\mathbb{Z}/2\mathbb{Z}$ homology balls. We also establish that every knot bounds a M{\"o}bius band in a rational homology ball, and that there are knots whose genus in $T^4$ and $B^4$ differ arbitrarily.

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Clayton McDonald, Allison N. Miller. 2025-11-19. Constructing knots with low rational genera. https://arxiv.org/abs/2511.15900

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