Bridge Multisections of Knotted Surfaces in $S^4$

TL;DR

This paper establishes a complete set of moves for relating different bridge multisections of surfaces in 4-space, introduces a band surgery operation, and shows how these techniques can unknot surfaces via finitely many surgeries.

Contribution

It proves a uniqueness theorem for bridge multisections, develops band surgery as a new operation, and demonstrates how any multisected surface can be unknotted through finitely many band surgeries.

Findings

Complete set of moves relating any two multiplane diagrams of the same surface.

Band surgery operation enables transformations of multiplane diagrams.

Any multisected surface in S^4 can be unknotted by finitely many band surgeries.

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.