On two types of $Z$-monodromy in triangulations of surfaces

TL;DR

This paper classifies two specific types of $z$-monodromies in surface triangulations and proves that the subgraphs formed by edges with these monodromies are forests, with applications to $z$-knotted triangulations.

Contribution

It introduces a classification of $z$-monodromies into two types and proves that the associated subgraphs are forests, providing new structural insights into surface triangulations.

Findings

Edges with $z$-monodromy of type (M1) form a forest.

Edges with $z$-monodromy of type (M2) form a forest.

Application to connected sum of $z$-knotted triangulations.

Abstract

Let $\Gamma$ be a triangulation of a connected closed $2$-dimensional (not necessarily orientable) surface. Using zigzags (closed left-right paths), for every face of $\Gamma$ we define the $z$-monodromy which acts on the oriented edges of this face. There are precisely $7$ types of $z$-monodromies. We consider the following two cases: (M1) the $z$-monodromy is identity, (M2) the $z$-monodromy is the consecutive passing of the oriented edges. Our main result is the following: the subgraphs of the dual graph $\Gamma^{*}$ formed by edges whose $z$-monodromies are of types (M1) and (M2), respectively, both are forests. We apply this statement to the connected sum of $z$-knotted triangulations.