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

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.