Locally Helical Surfaces have bounded twisting

TL;DR

This paper proves that the total net twisting of locally helical surfaces in a fixed triangulated 3-manifold is bounded, providing a key topological constraint on their structure.

Contribution

It establishes a bound on the net twisting of helical pieces in topologically minimal surfaces within triangulated 3-manifolds, advancing understanding of their geometric properties.

Findings

Net twisting of helical pieces is bounded in any fixed triangulated 3-manifold.

Normal forms of topologically minimal surfaces include helicoids with controlled twists.

Provides a topological constraint relevant for the classification of surfaces in 3-manifolds.

Abstract

A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and helicoids. Helical pieces can have any number of positive or negative twists. We show here that the net twisting of the helical pieces of any such surface in a given triangulated 3-manifold is bounded.