# Connected sums of $Z$-knotted triangulations

## TL;DR

This paper characterizes when the connected sum of two $z$-knotted triangulations results in a $z$-knotted triangulation, expanding understanding of zigzag properties in embedded graphs.

## Contribution

It provides a complete description of conditions under which the connected sum of two $z$-knotted triangulations remains $z$-knotted.

## Key findings

- Identifies all cases where connected sums preserve $z$-knottedness.
- Provides a classification of $z$-knotted triangulations under connected sum operations.
- Enhances understanding of zigzag structures in embedded graphs.

## Abstract

An embedded graph is called $z$-knotted if it contains the unique zigzag (up to reversing). We consider $z$-knotted triangulations, i.e. $z$-knotted embedded graphs whose faces are triangles, and describe all cases when the connected sum of two $z$-knotted triangulations is $z$-knotted.