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.

Let Γ be a triangulation of a connected closed 2-dimensional (not necessarily orientable) surface. Using zigzags (closed left-right paths), for every face of Γ 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 Γ * 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.

An embedded graph is called z-knotted if it contains a 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. To the memory of Michel Marie Deza