Z-knotted triangulations of surfaces

TL;DR

This paper proves that every triangulation of a closed surface can be transformed into a z-knotted map, which contains a single zigzag, revealing new structural properties of surface embeddings.

Contribution

It introduces the concept of z-knotted triangulations and demonstrates that any triangulation of a closed surface admits a z-knotted shredding, a novel structural result.

Findings

Every surface triangulation admits a z-knotted shredding.

Z-knotted maps are connected to the Gauss code problem.

Zigzags have notable homological properties.

Abstract

A zigzag in a map (a $2$-cell embedding of a connected graph in a connected closed $2$-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a common vertex, 2) for any three consecutive edges the first and the third edges are disjoint and the face containing the first and the second edges is distinct from the face which contains the second and the third. A map is $z$-knotted if it contains a single zigzag. Such maps are closely connected to Gauss code problem and have nice homological properties. We show that every triangulation of a connected closed $2$-dimensional surface admits a $z$-knotted shredding.