$Z$-knotted and $Z$-homogeneous triangulations of surfaces

TL;DR

This paper studies special triangulations of surfaces characterized by unique zigzag structures, introducing classifications like $z$-knotted and $z$-homogeneous, and presents an algorithm to transform such triangulations while preserving their properties.

Contribution

It defines new classes of triangulations based on zigzag configurations and provides an algorithm to generate related $z$-homogeneous triangulations.

Findings

Characterization of $z$-knotted and $z$-homogeneous triangulations.

Development of an algorithm to transform $z$-homogeneous triangulations.

Identification of properties linking zigzag types and triangulation structure.

Abstract

A triangulation is called $z$-knotted if it has a single zigzag (up to reversing). A $z$-orientation on a triangulation is a minimal collection of zigzags which double covers the set of edges. An edge is of type I if zigzags from the $z$-orientation pass through it in different directions, otherwise this edge is of type II. If all zigzags from the $z$-orientation contain precisely two edges of type I after any edge of type II, then the $z$-oriented triangulation is said to be $z$-homogeneous. We describe an algorithm transferring each $z$-homogeneous trianguation to other $z$-homogeneous triangulation which is also $z$-knotted.