Bridge Multisections of Knotted Surfaces in $S^4$

Román Aranda; Carolyn Engelhardt

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arXiv · 2410.01921

Bridge Multisections of Knotted Surfaces in $S^4$

Román Aranda·Carolyn Engelhardt

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Abstract

Bridge multisections are combinatorial descriptions of surface links in 4-space using tuples of trivial tangles. They were introduced by Islambouli, Karimi, Lambert-Cole, and Meier to study curves in rational surfaces. In this paper, we prove a uniqueness result for bridge multisections of surfaces in 4-space: we give a complete set of moves relating to any two multiplane diagrams of the same surface. This is done by developing a surgery operation on multiplane diagrams called band surgery. Another application of this surgery move is that any $n$-valent graph with an $n$-edge coloring is the spine of a bridge multisection for an unknotted surface. We also prove that any multisected surface in $S^4$ can be unknotted by finitely many band surgeries.

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BibTeX RIS

Román Aranda, Carolyn Engelhardt. 2026-04-10. Bridge Multisections of Knotted Surfaces in $S^4$. https://arxiv.org/abs/2410.01921

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