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

Region colorings of surfaces in 4-space

Geometric Topology 2025-04-22 v1

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

View on arXiv ↗ PDF ↗

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 $F\not=-F$ .

Keywords

region colorings surface links triplane diagrams knot theory non-invertible knots

Cite

@article{arxiv.2504.14770,
  title  = {Region colorings of surfaces in 4-space},
  author = {Román Aranda and Noah Crawford and Andrew Maas and Nicole Marienau and Erica Pearce and Renzo Sarreal and Savannah Schutte and Ransom Sterns and Eric Woods},
  journal= {arXiv preprint arXiv:2504.14770},
  year   = {2025}
}

Comments

20 pages, 23 figures, comments are welcome

Region colorings of surfaces in 4-space

Abstract

Keywords

Cite

Comments

Related papers