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.

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.

Equations

$$a,1,2,b,3,4,\dots ,a,n-1,n,$$

$$b,1,2,a,3,4,\dots ,b,n-1,n$$

$$a,1,2,b,3,4,a,\dots ,b,n-1,n$$

$$a,1,2,b,3,1,a,2,3,b,1,2,a,3,1,b,2,3$$

$$a,1,2,b,3,4,\dots ,a,n-2,n-1,b,n,1,a,2,3,b,4,5,\dots ,a,n-1,n,$$

$$b,1,2,a,3,4,\dots ,b,n-2,n-1,a,n,1,b,2,3,a,4,5,\dots ,b,n-1,n$$

$$a,1,2,b,3,4,a,5,1,b,2,3,a,4,5,$$

$$b,1,2,a,3,4,b,5,1,a,2,3,b,4,5$$

$$a,1,2,b,3,4,a,\dots ,b,n-2,n-1,a,n,1,b,2,3,a,\dots ,a,n-1,n,$$

$$b,1,2,a,3,4,b,\dots ,a,n-2,n-1,b,n,1,a,2,3,b,\dots ,b,n-1,n$$

$$\dots ,x,y,z,\dots ,y,x,z,\dots ,y,z,x,\dots$$

$$\dots ,x,y,z,\dots ,y,z,x,\dots ,y,x,z,\dots$$

$$\dots ,p,x,y,p^{\prime }\dots ,q,y,x,q^{\prime }\dots ,$$

$$x,y,z,\dots ,\delta (x),\delta (y),\delta (z),\dots ,\gamma (x),\gamma (y),\gamma (z),\dots ,$$

$$A=\{y,z,\dots ,\delta (x),\delta (y)\},$$

$$B=\{\delta (y),\delta (z),\dots ,\gamma (x),\gamma (y)\},$$

$$C=\{\gamma (y),\gamma (z)\dots ,x,y\},$$

$$x^{\prime },y^{\prime },z^{\prime },\dots ,{\delta }^{\prime }(x^{\prime }),{\delta }^{\prime }(y^{\prime }),{\delta }^{\prime }(z^{\prime }),\dots ,{\gamma }^{\prime }(x^{\prime }),{\gamma }^{\prime }(y^{\prime }),{\gamma }^{\prime }(z^{\prime }),\dots ,$$

$$A^{\prime }=\{y^{\prime },z^{\prime },\dots ,{\delta }^{\prime }(x^{\prime }),{\delta }^{\prime }(y^{\prime })\},$$

$$B^{\prime }=\{{\delta }^{\prime }(y^{\prime }),{\delta }^{\prime }(z^{\prime }),\dots ,{\gamma }^{\prime }(x^{\prime }),{\gamma }^{\prime }(y^{\prime })\},$$

$$C^{\prime }=\{{\gamma }^{\prime }(y^{\prime }),{\gamma }^{\prime }(z^{\prime })\dots ,x^{\prime },y^{\prime }\}.$$

$$X^{\prime }\in \{A^{\prime },B^{\prime },C^{\prime },A^{\prime -1},B^{\prime -1},C^{\prime -1}\}$$

$$Y\in \{A,B,C,A^{-1},B^{-1},C^{-1}\}\setminus \{X,X^{-1}\}$$

$$Y^{\prime }\in \{A^{\prime },B^{\prime },C^{\prime },A^{\prime -1},B^{\prime -1},C^{\prime -1}\}\setminus \{X^{\prime },X^{\prime -1}\}.$$

$$X,X^{\prime }\text{mbox}orX,X^{\prime },Y,Y^{\prime }\text{mbox}orX,X^{\prime },Y,Y^{\prime },Z,Z^{\prime },$$

$$\{A,B,C,A^{-1},B^{-1},C^{-1}\}\text{mbox}and\{A^{\prime },B^{\prime },C^{\prime },A^{\prime -1},B^{\prime -1},C^{\prime -1}\},$$

$$a,1,2,\dots ,1,a,2,\dots ,1,2,a,\dots$$

$$a^{\prime },1^{\prime },2^{\prime },\dots ,1^{\prime },a^{\prime },2^{\prime },\dots ,1^{\prime },2^{\prime },a^{\prime },\dots$$

$$A=\{1,2,\dots ,1,a\},B=\{a,2,\dots ,1,2\},C=\{2,a,\dots ,a,1\},$$

$$A^{\prime }=\{1^{\prime },2^{\prime },\dots ,1^{\prime },a^{\prime }\},B^{\prime }=\{a^{\prime },2^{\prime },\dots ,1^{\prime },2^{\prime }\},C^{\prime }=\{2^{\prime },a^{\prime },\dots ,a^{\prime },1^{\prime }\}.$$

Full text

To the memory of Michel Marie Deza

1. Introduction

The notion of Petrie polygon for polytopes was introduced in Coxeter’s book [1] (see also [4, 6, 11] and [5, Chapter 8] for some generalizations). For graphs embedded in $2$-dimensional surfaces the same objects appear as zigzags in [2, 3, 5], geodesics in [7] and left-right paths in [10]. Following [2, 3, 4, 5] we call them zigzags.

A zigzag is a cyclic sequence of vertices in an embedded graph $\Gamma$, where any two consecutive vertices form an edge, any three consecutive vertices are mutually distinct vertices on a face and any four consecutive vertices are not on the same face. For every zigzag the reversed sequence also is a zigzag and we do not distinguish such two zigzags. When there is a unique zigzag (up to reversing), we say that $\Gamma$ is $z$-knotted.

The notion of zigzag is closely connected to the notion of central circuit in a $4$-regular embedded graph. Zigzags and central circuits are interesting for many reasons. For example, there is an application to knot theory related to projections of links [5, Section 1.3]. In particular, $z$-knotted embedded graphs connect to the Gauss code problem as follows. The medial graph of such a graph has a unique central circuit, and this medial graph can be considered as a closed curve having simple self-intersections only, i.e. it is a realization of a certain Gauss code on the corresponding surface (see [9]).

The main objects of this paper are $z$-knotted triangulations ($z$-knotted embedded graphs whose faces are triangles). Note that the dual objects, i.e. $z$-knotted $3$-regular graphs, are considered in [5, Section 2.3]. A simple example of $z$-knotted triangulation is the graph of the $n$-gonal bipyramid when $n$ is odd (this graph is not $z$-knotted if $n$ is even). Examples of $z$-knotted fullerenes can be found in [5, Section 2.3] (the dual graphs are triangulations). In this paper, we show how construct more examples.

The connected sum of manifolds is a well-known topological notion. In a similar way, we define the connected sum of triangulations $\Gamma$ and $\Gamma^{\prime}$ as the gluing by two distinguished faces $F$ and $F^{\prime}$ (in $\Gamma$ and $\Gamma^{\prime}$, respectively). This is a triangulation in the connected sum of the surfaces containing $\Gamma$ and $\Gamma^{\prime}$, and it depends on an identification homeomorphism between $\partial F$ and $\partial F^{\prime}$. We describe all cases when the connected sum of two $z$-knotted triangulations is $z$-knotted.

2. Zigzags in embedded graphs

Let $M$ be a connected closed $2$-dimensional surface (not necessarily orientable). Let also $\Gamma$ be a finite simple connected graph embedded in $M$ such that every vertex of $\Gamma$ is a point on $M$, every edge is homeomorphic to the segment $[0;1]$ and any two edges are disjoint or intersecting in a vertex. The faces of $\Gamma$ are defined as the closures of the connected components of $M\setminus\Gamma$. We suppose that our graph embedding is a closed $2$-cell embedding, i.e. every face is homeomorphic to a closed $2$-dimensional disc. Also, we require that the following two conditions hold:

1. $\bullet$

every edge is contained in precisely two distinct faces, 2. $\bullet$

the intersection of two distinct faces is an edge or a vertex or empty.

A zigzag in $\Gamma$ is a sequence of vertices $\{x_{i}\}_{i\in{\mathbb{N}}}$ satisfying the following three conditions for every $i\in{\mathbb{N}}$:

1. (Z1)

$x_{i},x_{i+1}$ are distinct vertices on a certain edge, 2. (Z2)

there is a unique face containing $x_{i},x_{i+1},x_{i+2}$, 3. (Z3)

the faces containing $x_{i},x_{i+1},x_{i+2}$ and $x_{i+1},x_{i+2},x_{i+3}$ are distinct.

Let $Z=\{x_{i}\}_{i\in{\mathbb{N}}}$ be a zigzag in $\Gamma$. Since our graph is finite, there is a natural number $n>3$ such that $x_{i+n}=x_{n}$ for every $i\in{\mathbb{N}}$. The smallest number $n$ satisfying this condition is called the length of $Z$. So, our zigzag is the cyclic sequence $x_{1},\dots,x_{n}$, where $n$ is the length of $Z$. For every natural $j$ the sequence $\{x_{i+j}\}_{i\in{\mathbb{N}}}$ is a zigzag, and all such zigzags are identified with $Z$.

Suppose that $x_{1},x_{2},x_{3}$ are three distinct vertices on a certain face and $x_{2}$ is adjacent to both $x_{1},x_{3}$. Consider the second face containing $x_{2},x_{3}$. It contains the unique vertex $x_{4}$ adjacent to $x_{3}$ and distinct from $x_{2}$. We apply the same arguments to $x_{2},x_{3},x_{4}$ and get a vertex $x_{5}$. Step by step, we construct a sequence $\{x_{i}\}_{i\in{\mathbb{N}}}$ satisfying the conditions (Z1) – (Z3). Therefore, every zigzag $\{x_{i}\}_{i\in{\mathbb{N}}}$ is completely determined by any three consecutive vertices $x_{i},x_{i+1},x_{i+2}$.

If $X=\{x_{1},\dots,x_{n}\}$ is a sequence of vertices in $\Gamma$, then $X^{-1}$ denotes the reversed sequence $x_{n},\dots,x_{1}$. If $Z$ is a zigzag, then the same holds for $Z^{-1}$. If $Z$ contains a sequence $x,y,z$, then the sequence $z,y,x$ is contained the reversed zigzag $Z^{-1}$. A sequence of type $x,y,z,\dots,z,y,x,\dots$ cannot be a zigzag (see, for example, [6]), i.e. a zigzag is not reversed to itself. Every zigzag will be identified with its reverse.

Example 1**.**

See Fig.1 for examples of zigzags in the complete graph $K_{4}$ and the cube graph $Q_{3}$.

For a zigzag $\{x_{i}\}_{i\in{\mathbb{N}}}$ in $\Gamma$ the sequence $\{F_{i}\}_{i\in{\mathbb{N}}}$, where $F_{i}$ is the face containing $x_{i},x_{i+1},x_{i+2}$, is a zigzag in the dual graph $\Gamma^{*}$. There is a one-to-one correspondence between zigzags in $\Gamma$ and $\Gamma^{*}$.

3. $z$-knotted graphs

The graph $\Gamma$ is called $z$-knotted if it contains the unique zigzag (up to reversing).

Let $e$ be an edge in $\Gamma$ and let $x,y$ be the vertices on this edge. We take one of the two faces containing $e$. This face contains a unique vertex $p\neq y$ adjacent to $x$ and the unique vertex $q\neq x$ adjacent to $y$ (if $p=q$, then the face is a triangle). The sequences $p,x,y$ and $x,y,q$ define some zigzags (not necessarily distinct) and the reversed zigzags correspond to the sequences $y,x,p$ and $q,y,x$, respectively. See Fig.2. If $\Gamma$ is $z$-knotted, then the zigzags determined by $p,x,y$ and $x,y,q$ must coincide (up to reversal), and so the unique zigzag contains, say, $p,x,y$ and either $x,y,q$ or $q,y,x$, i.e. one of the following possibilities is realized:

1. (1)

the unique zigzag is a sequence $\dots,x,y,\dots,y,x,\dots$, 2. (2)

the unique zigzag is a sequence $\dots,x,y\dots,x,y,\dots$.

The edge $e$ is said to be of first or of second type if the corresponding case is realized. So, the zigzag in a $z$-knotted graph passes through every edge twice in different directions (first type) or in the same direction (second type). Conversely, if there is a zigzag of $\Gamma$ passing through every edge twice, then $\Gamma$ is $z$-knotted.

Example 2**.**

Consider the graph of the $n$-gonal bipyramid $BP_{n}$, $n\geq 3$ which consists of an $n$-gone whose vertices are denoted by $1,\dots,n$ and connected with two disjoint vertices $a,b$. Suppose that $n=2k$. If $k$ is odd, then the sequence

[TABLE]

[TABLE]

is a zigzag. In the case when $k$ is even, one of the zigzags is $a,1,2,b,3,4$ if $k=2$ and

[TABLE]

if $k\geq 4$. Each of these sequences passes through the edge joining $a$ and $1$ only ones. This means that $BP_{n}$ is not $z$-knotted if $n$ is even. Consider the case when $n$ is odd, i.e. $n=2k+1$ and $k\geq 1$. If $k=1$, then

[TABLE]

is a zigzag. Similarly, if $k$ is odd and not less than $3$, then

[TABLE]

[TABLE]

is a zigzag. In the case when $k=2$, the sequence

[TABLE]

[TABLE]

is a zigzag. If $k$ is even and not less than $4$, then

[TABLE]

[TABLE]

is a zigzag. The length of these zigzags is $6n$. Since $BP_{n}$ contains precisely $3n$ edges, each of these zigzags passes through all edges twice. This guarantees that there are no other zigzags (except the reverse) and $BP_{n}$ is $z$-knotted if $n$ is odd. Every edge joining $x\in\{a,b\}$ and $y\in\{1,\dots,n\}$ is of first type and all edges in the $n$-gon are of second type. Each face of $BP_{n}$ is a triangle formed by two edges of first type and one edge of second type. The zigzag passes through each face thrice. Suppose that $x\in\{a,b\}$ and $y,z\in\{1,\dots,n\}$ are the vertices on a certain face. An immediate verification shows that the zigzag is a sequence

[TABLE]

if $k$ is odd and it is a sequence

[TABLE]

if $k$ is even.

Suppose that $\Gamma$ is $z$-knotted. This is equivalent to the dual graph $\Gamma^{*}$ being $z$-knotted. If $e$ is an edge of first type in $\Gamma$, then the zigzag is a sequence

[TABLE]

see Fig.2. Denote by $F$ and $F^{\prime}$ the faces containing $p,x,y,q$ and $p^{\prime},x,y,q^{\prime}$, respectively. The edge $e$ corresponds to the edge of $\Gamma^{*}$ joining $F$ and $F^{\prime}$. The zigzag in $\Gamma^{*}$ is a sequence $\dots,F,F^{\prime},\dots,F,F^{\prime},\dots$, i.e. the edge of $\Gamma^{*}$ corresponding to $e$ is of second type.

Similarly, every edge of second type in $\Gamma$ corresponds to an edge of first type in $\Gamma^{*}$.

4. Two types of faces in $z$-knotted triangulations

In this section, we suppose that $\Gamma$ is $z$-knotted and every face of $\Gamma$ is a triangle, i.e. $\Gamma$ is a $z$-knotted triangulation.

Lemma 1**.**

For every face $F$ one of the following possibilities is realized:

1. (1)

only one edge in $F$ is of second type, 2. (2)

all edges in $F$ are of second type.

In the cases (1) and (2), we say that $F$ is a $(1,1,2)$-face or a $(2,2,2)$-face, respectively.

Proof of Lemma 1.

Observe that the zigzag passes through the face $F$ thrice; i.e. there are precisely three intersections of the zigzag with $F$ which contain more than one vertex. Every such intersection consists of two edges.

This observation shows that the following possibilities cannot be realized: $F$ contains precisely two edges of second type (see Fig.3 (a) and (b)) and all edges in $F$ are of first type (see Fig.3 (c)). ∎

Suppose that $F$ is a $(1,1,2)$-face. Let $x,y,z$ be the vertices of $F$. We assume that the vertices $y,z$ are on a unique edge of second type and the zigzag goes twice from $y$ to $z$, see Fig.4.

Then the zigzag contains the sequences $x,y,z$ and $y,z,x$. The zigzag passes through $F$ thrice and the third sequence formed by the vertices of $F$ is $y,x,z$. Since the zigzag is a cyclic sequence, we can suppose $x,y,z$ are the first three consecutive vertices in the zigzag. The remaining two sequences associated to $F$ are $y,z,x$ and $y,x,z$. There are precisely the following two possibilities for the zigzag:

1. (od)

$x,y,z,\dots,y,x,z,\dots,y,z,x,\dots$, 2. (ev)

$x,y,z,\dots,y,z,x,\dots,y,x,z,\dots$.

The first case is realized for all faces in the bipyramid graph $BP_{2k+1}$ if $k$ is odd. In this case, we say that $F$ is a $(1,1,2)$-face of odd type. The second case corresponds to all faces in $BP_{2k+1}$ if $k$ is even, and $F$ is said to be a $(1,1,2)$-face of even type in this case.

Let $F$ be a $(2,2,2)$-face and let $x,y,z$ be the vertices on $F$. Formally, we have two possibilities (Fig.5); but the second cannot happen because the zigzag passes through each face thrice. So, the zigzag defines the orientation on $F$. We suppose that this orientation coincides with the order of vertices in the sequence $x,y,z$ (the first triangle on Fig.5).

The zigzag contains the sequences $x,y,z$ and $y,z,x$ and $z,x,y$. Since the zigzag is a cyclic sequence, we can suppose that the first three consecutive vertices are $x,y,z$. Then there are only the following two possibilities for the zigzag:

1. (1)

$x,y,z,\dots,z,x,y,\dots,y,z,x,\dots$, 2. (2)

$x,y,z,\dots,y,z,x,\dots,z,x,y,\dots$.

We say that $F$ is a $(2,2,2)$-face of first or of second type if the corresponding case is realized. See Examples 4 and 5 for $z$-knotted triangulations containing $(2,2,2)$-faces of first type. In Section 7, we construct a $z$-knotted triangulation with a $(2,2,2)$-face of second type.

5. Main result

Let $\Gamma$ and $\Gamma^{\prime}$ be triangulations of closed connected $2$-dimensional surfaces $M$ and $M^{\prime}$, respectively. Suppose that $F$ is a face in $\Gamma$ and $F^{\prime}$ is a face in $\Gamma^{\prime}$. These faces both are homeomorphic to a closed $2$-dimensional disc and each of the boundaries $\partial F$ and $\partial F^{\prime}$ is the sum of three edges. Let $g:\partial F\to\partial F^{\prime}$ be a homeomorphism transferring every vertex of $F$ to a vertex of $F^{\prime}$, i.e. if $x_{i}$, $i\in\{1,2,3\}$ are the vertices of $F$, then $x^{\prime}_{i}=g(x_{i})$, $i\in\{1,2,3\}$ are the vertices of $F^{\prime}$. The connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is the graph whose vertex set is the union of the vertex sets of $\Gamma$ and $\Gamma^{\prime}$, where every $x_{i}$ is identified with $x^{\prime}_{i}$, and the edge set is the union of the edge sets of $\Gamma$ and $\Gamma^{\prime}$, where the edge containing $x_{i},x_{j}$ is identified with the edge containing $x^{\prime}_{i},x^{\prime}_{j}$. This is a triangulation in the connected sum of $M$ and $M^{\prime}$.

Example 3**.**

Any connected sum of two exemplars of $K_{4}$ is the bipyramid graph $BP_{3}$. One of the connected sums of two exemplars of $BP_{3}$ is presented on Fig.6.

Later, we show that it is a $z$-knotted triangulation containing $(2,2,2)$-faces of first type.

Theorem 1**.**

Suppose that $\Gamma$ and $\Gamma^{\prime}$ are $z$-knotted. Then the following assertions are fulfilled:

1. (1)

If $F$ and $F^{\prime}$ are $(1,1,2)$-faces in $\Gamma$ and $\Gamma^{\prime}$ (respectively), then there is a homeomorphism $g:\partial F\to\partial F^{\prime}$ transferring vertices to vertices and such that the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted. 2. (2)

If $F$ is a $(2,2,2)$-face of first type in $\Gamma$, then for every face $F^{\prime}$ in $\Gamma^{\prime}$ and every homeomorphism $g:\partial F\to\partial F^{\prime}$ transferring vertices to vertices the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted. 3. (3)

Suppose that $F$ is a $(2,2,2)$-face of second type in $\Gamma$ and $F^{\prime}$ is a face in $\Gamma^{\prime}$ such that the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted for a certain homeomorphism $g:\partial F\to\partial F^{\prime}$ transferring vertices to vertices. Then $F^{\prime}$ is a $(2,2,2)$-face of first type or a $(1,1,2)$-face of odd type. In these cases, $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted for every homeomorphism $g:\partial F\to\partial F^{\prime}$ transferring vertices to vertices.

Using this result we construct the following examples of $z$-knotted triangulations containing $(2,2,2)$-faces of first type:

1. $\bullet$

If $F$ is a $(1,1,2)$-face of odd type in a $z$-knotted triangulation $\Gamma$ and $F^{\prime}$ is a face in the bipyramid graph $BP_{2k+1}$, where $k$ is odd, then for some homeomorphisms $g:\partial F\to\partial F^{\prime}$ transferring vertices to vertices the connected sum $\Gamma\#_{g}BP_{2k+1}$ is $z$-knotted and contains $(2,2,2)$-faces of first type (Example 4). 2. $\bullet$

If $F$ is a $(1,1,2)$-face of even type in a $z$-knotted triangulation $\Gamma$ and $F^{\prime}$ is a face in the bipyramid graph $BP_{2k+1}$, where $k$ is even, then for some homeomorphisms $g:\partial F\to\partial F^{\prime}$ transferring vertices to vertices the connected sum $\Gamma\#_{g}BP_{2k+1}$ is $z$-knotted and contains $(2,2,2)$-faces of first type (Example 5).

Remark 1**.**

Examples of $z$-knotted fullerenes can be found in [5, p.31, Fig 2.2]. The dual graphs are $z$-knotted triangulations whose vertices are of degree $5$ or $6$. Observe that each vertex of degree $5$ is adjacent to every vertex of a certain $5$-gone, but there is no other vertex adjacent to all vertices of this $5$-gone. This means that these triangulations cannot be presented as the connected sums of some collections of $z$-knotted bipyramids.

Remark 2**.**

If $E$ is the edge set of a $z$-knotted embedded graph, then the cyclic ordering of $E$ (associated to the zigzag) induces two linear transformations of $2^{E}$ (considered as a vector space over ${\mathbb{F}}_{2}$) [8]. It will be interesting to describe relations between such transformations for $\Gamma,\Gamma^{\prime}$ and $\Gamma\#_{g}\Gamma^{\prime}$ (if this connected sum is $z$-knotted).

6. Proof of Theorem 1

6.1. General construction of zigzags in the connected sum

From this moment, we will suppose that $\Gamma$ and $\Gamma^{\prime}$ are $z$-knotted triangulations. Let $F$ and $F^{\prime}$ be faces in $\Gamma$ and $\Gamma^{\prime}$, respectively. If $x,y,z$ are the vertices of $F$, then the zigzag of $\Gamma$ is a sequence

[TABLE]

where $\delta$ and $\gamma$ are permutations on the set $\{x,y,z\}$. Let us consider this zigzag as the union of the following three segments

[TABLE]

[TABLE]

[TABLE]

i.e. as the cyclic sequence $A,B,C$. It must be pointed out that any two consecutive segments have the same vertex (for example, the segments $A$ and $B$ are joined in the vertex $\delta(y)$). Similarly, if $x^{\prime},y^{\prime},z^{\prime}$ are the vertices of $F^{\prime}$, then the zigzag of $\Gamma^{\prime}$ is a sequence

[TABLE]

where $\delta^{\prime}$ and $\gamma^{\prime}$ are permutations on the set $\{x^{\prime},y^{\prime},z^{\prime}\}$. We consider this zigzag as the cyclic sequence $A^{\prime},B^{\prime},C^{\prime}$, where

[TABLE]

[TABLE]

[TABLE]

Let $g:\partial F\to\partial F^{\prime}$ be a homeomorphism transferring vertices to vertices. Zigzags in the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ can be constructed as follows. We start from a segment $X\in\{A,B,C\}$. Let $a,b\in\{x,y,z\}$ be the last two vertices in $X$ (the order of the vertices is important). There is a unique segment

[TABLE]

such that $g(a),g(b)\in\{x^{\prime},y^{\prime},z^{\prime}\}$ are the first two vertices in $X^{\prime}$ (as above, the order of the vertices is important). Indeed, if $c^{\prime}$ is the element of $\{x^{\prime},y^{\prime},z^{\prime}\}$ distinct from $g(a),g(b)$ and there are two segments satisfying this condition, then the zigzag in $\Gamma^{\prime}$ contains the sequence $g(a),g(b),c^{\prime}$ twice or it contains this sequence together with the reversed sequence $c^{\prime},g(b),g(a)$; each of these cases is impossible.

Let $a^{\prime},b^{\prime}\in\{x^{\prime},y^{\prime},z^{\prime}\}$ be the last two vertices in $X^{\prime}$. There is a unique segment

[TABLE]

such that $g^{-1}(a^{\prime}),g^{-1}(b^{\prime})$ are the first two vertices in $Y$. If $Y=X$, then $X,X^{\prime}$ is a zigzag in $\Gamma\#_{g}\Gamma^{\prime}$. In the case when $X$ and $Y$ are distinct, we apply the same arguments to $Y$ and, as above, choose a certain segment

[TABLE]

Step by step, we construct a cyclic sequence which defines a zigzag in $\Gamma\#_{g}\Gamma^{\prime}$. The intersection of any two consecutive segments in this sequence consists of precisely two vertices. There are the following three possibilities for the associated zigzag:

[TABLE]

where $X,Y,Z$ and $X^{\prime},Y^{\prime},Z^{\prime}$ are distinct elements of

[TABLE]

respectively. In the third case, the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted. Otherwise, $\Gamma\#_{g}\Gamma^{\prime}$ contains three zigzags of type $X,X^{\prime}$ or one zigzag of type $X,X^{\prime}$ and one zigzag of type $Y,Y^{\prime},Z,Z^{\prime}$.

6.2. Proof of the statement (1)

In this subsection, we suppose that $F$ and $F^{\prime}$ both are $(1,1,2)$-faces, i.e. similar to faces in the $z$-knotted bipyramids. As in Example 2, we denote by $a$ and $a^{\prime}$ the vertices of $F$ and $F^{\prime}$ (respectively) which do not belong to the edges of second type. Also, we write $1,2$ and $1^{\prime},2^{\prime}$ for the vertices on the edge of second type in $F$ and $F^{\prime}$, respectively. We will assume that the zigzags in $\Gamma$ and $\Gamma^{\prime}$ go from $1$ and $1^{\prime}$ to $2$ and $2^{\prime}$, respectively.

Case 1 (${\rm od}+{\rm od}$). If $F$ and $F^{\prime}$ are of odd type, then the zigzags in $\Gamma$ and $\Gamma^{\prime}$ are sequences

[TABLE]

and

[TABLE]

(respectively) and we have

[TABLE]

[TABLE]

There are precisely $6$ distinct bijections between the vertex sets of $F$ and $F^{\prime}$. In the case when $a$ is identified with $a^{\prime}$, the corresponding connected sums are $z$-knotted. If $1,2$ are identified with $1^{\prime},2^{\prime}$ (respectively), then the unique zigzag in the connected sum is

[TABLE]

If $1$ is identified with $2^{\prime}$ and $2$ is identified with $1^{\prime}$, then this is

[TABLE]

In the remaining four cases, the connected sum contains precisely two zigzags and is not $z$-knotted (Tab.1).

Example 4**.**

Suppose that $\Gamma$ is the $n$-bipyramid graph considered in Example 2, $n=2k+1$ and $k$ is odd. Then

[TABLE]

[TABLE]

If $a,1,2$ are identified with $a^{\prime},1^{\prime},2^{\prime}$ (respectively), then the unique zigzag in the connected sum is $A,C^{\prime-1},B,A^{\prime},C^{-1},B^{\prime}$. The face containing $b,1,2$ appears in this zigzag as follows

[TABLE]

which means that it is a $(2,2,2)$-face of first type. The same holds for the case when $a,1,2$ are identified with $a^{\prime},2^{\prime},1^{\prime}$, respectively.

Case 2 (${\rm ev}+{\rm ev}$). If $F$ and $F^{\prime}$ are of even type, then the zigzags in $\Gamma$ and $\Gamma^{\prime}$ are sequences

[TABLE]

and

[TABLE]

(respectively) and

[TABLE]

[TABLE]

In contrast to the previous case, the connected sum is $z$-knotted if $a$ is not identified with $a^{\prime}$ (Tab.2).

In the case when $a$ is identified with $a^{\prime}$, the connected sum contains precisely three zigzags and is not $z$-knotted. If $1,2$ is identified with $1^{\prime},2^{\prime}$ (respectively), then these zigzags are $A,A^{\prime}$ and $B,C^{\prime-1}$ and $C,B^{\prime-1}$. If $1$ is identified with $2^{\prime}$ and $2$ is identified with $1^{\prime}$, then we get the zigzags $A,A^{\prime-1}$ and $B,B^{\prime}$ and $C,C^{\prime}$.

Example 5**.**

Suppose that $\Gamma$ is the $n$-bipyramid graph considered in Example 2, $n=2k+1$ and $k$ is even. Then

[TABLE]

[TABLE]

If $a,1,2$ are identified with $1^{\prime},2^{\prime},a^{\prime}$ (respectively), then the unique zigzag in the connected sum is $A,B^{\prime},C,A^{\prime},B^{-1},C^{\prime-1}$. The face containing $b,n-1,n$ appears in this zigzag as follows

[TABLE]

i.e. this is a $(2,2,2)$-face of first type. The same holds for the other three cases when $a$ is not identified with $a^{\prime}$.

Case 3 (${\rm ev}+{\rm od}$). If $F$ is of even type and $F^{\prime}$ is of odd type, then

[TABLE]

[TABLE]

The connected sum is $z$-knotted if $a$ is identified with $a^{\prime}$ (Tab.3).

For each of the remaining four cases, the connected sum contains precisely two zigzags and is not $z$-knotted (Tab.4).

Remark 3**.**

Suppose that $\Gamma$ and $\Gamma^{\prime}$ are the $n$-bipyramid and $n^{\prime}$-bipyramid graphs (respectively), where $n=2k+1$, $n^{\prime}=2k^{\prime}+1$, $k$ is odd and $k^{\prime}$ is even. Then all faces in any $z$-knotted connected sum of $\Gamma$ and $\Gamma^{\prime}$ are $(1,1,2)$-faces.

6.3. Proof of the statement (2)

In this subsection, we suppose that $F$ is a $(2,2,2)$-face of first type. We denote by $a,b,c$ the vertices of $F$ and assume that the orientation on $F$ defined by the zigzag coincides with the order of vertices in the sequence $a,b,c$. Then the zigzag is a sequence

[TABLE]

and we have

[TABLE]

We show that for every homeomorphism $g:\partial F\to\partial F^{\prime}$ the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted.

Consider the case when $F^{\prime}$ is a $(1,1,2)$-face. As in the previous subsection, we assume that $a^{\prime}$ is the vertex which do not belong to the edges of second type, $1^{\prime}$ and $2^{\prime}$ are the vertices on the edge of second type and the zigzag goes from $1^{\prime}$ to $2^{\prime}$. If $F^{\prime}$ is of odd type, then

[TABLE]

and the zigzags in the corresponding connected sums are presented in Tab.5.

In the case when $F^{\prime}$ is of even type, we have

[TABLE]

and the zigzags in the connected sums can be found in Tab.6.

Suppose that $F^{\prime}$ is a $(2,2,2)$-face whose vertices are $a^{\prime},b^{\prime},c^{\prime}$ and the orientation coincides with the order of vertices in the sequence $a^{\prime},b^{\prime},c^{\prime}$. If $F^{\prime}$ is of first type, then

[TABLE]

and the zigzags in the connected sums are described in Tab.7.

If $F^{\prime}$ is of second type, then the zigzag in $\Gamma^{\prime}$ is a sequence

[TABLE]

and

[TABLE]

The zigzags in the connected sums are presented in Tab.8.

6.4. Proof of the statement (3)

In this subsection, we suppose that $F$ is a $(2,2,2)$-face of second type. As in the previous subsection, we denote by $a,b,c$ the vertices of $F$ and assume that the orientation on $F$ defined by the zigzag coincides with the order of vertices in the sequence $a,b,c$. Then

[TABLE]

If $F^{\prime}$ is a $(2,2,2)$-face of first type, then the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted for any homeomorphism $g:\partial F\to\partial F^{\prime}$ (see the previous subsection). If $F^{\prime}$ is a $(2,2,2)$-face of second type, then

[TABLE]

In this case, for every of the segments $A,B,C,A^{\prime},B^{\prime},C^{\prime}$ the first two vertices coincides with the last two vertices. Therefore, for any homeomorphism $g:\partial F\to\partial F^{\prime}$ all zigzags in the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ are of type $X,X^{\prime}$, where

[TABLE]

So, the connected sum is not $z$-knotted.

Consider the case when $F^{\prime}$ is a $(1,1,2)$-face whose vertices are $a^{\prime},1^{\prime},2^{\prime}$ and $a^{\prime}$ is the vertex which does not belong to the edge of second type. As above, we suppose that the zigzag goes from $1^{\prime}$ to $2^{\prime}$. If $F^{\prime}$ is of odd type, then

[TABLE]

and for every homeomorphism $g:\partial F\to\partial F^{\prime}$ the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is $z$-knotted (Tab. 9).

If $F^{\prime}$ is of even type, then

[TABLE]

and for every homeomorphism $g:\partial F\to\partial F^{\prime}$ the connected sum $\Gamma\#_{g}\Gamma^{\prime}$ is not $z$-knotted (Tab.10).

7. $(2,2,2)$-faces of second type

Consider the bipyramid graph $BP_{n}$, where $n=2k$ and $k$ is odd. As in Example 2, we denote by $1,\dots,n$ the vertices on the $n$-gone and write $a,b$ for the remaining two vertices. There precisely two zigzags in $BP_{n}$. These are the sequences

[TABLE]

and

[TABLE]

Let $F$ be the face containing $a,1,2$. Following Subsection 6.1, we present zigzags as the unions of segments which are parts of zigzags between two edges of $F$, i.e. each of these segments contains precisely two edges of $F$. The first zigzag is the sum of the following two segments

[TABLE]

The second zigzag passes ones through the edges $a1$ and $a2$, and it does not contain the edge $12$. So, it will be identified with the segment

[TABLE]

Now, we take the bipyramid graph $BP_{n^{\prime}}$, where $n^{\prime}=2k^{\prime}$ and $k^{\prime}$ is odd. Let $1^{\prime},\dots,n^{\prime}$ be the vertices on the $n^{\prime}$-gone, and let $a^{\prime},b^{\prime}$ be the remaining two vertices. Denote by $F^{\prime}$ the face containing $a^{\prime},1^{\prime},2^{\prime}$. As above, one of the zigzags is the union of two segments $A^{\prime},B^{\prime}$ and the other zigzag is identified with the segment $C^{\prime}$.

We identify the vertices $a,1,2$ with the vertices $2^{\prime},a^{\prime},1^{\prime}$ (respectively). The corresponding connected sum of $BP_{n}$ and $BP_{n^{\prime}}$ contains the unique zigzag

[TABLE]

i.e. it is $z$-knotted. The face containing $a,2,3$ appears in this zigzag as follows

[TABLE]

This is a $(2,2,2)$-face of second type.