Region colorings of surfaces in 4-space

Román Aranda; Noah Crawford; Andrew Maas; Nicole Marienau; Erica Pearce; Renzo Sarreal; Savannah Schutte; Ransom Sterns; Eric Woods

arXiv · 2504.14770

Region colorings of surfaces in 4-space

Román Aranda·Noah Crawford·Andrew Maas·Nicole Marienau·Erica Pearce·Renzo Sarreal·Savannah Schutte·Ransom Sterns·Eric Woods

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Abstract

Niebrzydowski introduced a theory of region colorings for surface links. In this paper, we translate the coloring invariant to the context of triplane diagrams and movies of knots. We provide inequalities between the number of region colorings and topological quantities of $F$, such as the number of saddles in a movie and the bridge index of a triplane diagram of $F$. As an application, we show that Yoshikawa's 2-knots $9_1$ and $10_2$ are non-invertible; that is $F\not=-F$.

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Román Aranda, Noah Crawford, Andrew Maas, Nicole Marienau, Erica Pearce, Renzo Sarreal, Savannah Schutte, Ransom Sterns, Eric Woods. 2025-04-20. Region colorings of surfaces in 4-space. https://arxiv.org/abs/2504.14770

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