Z-knotted triangulations of surfaces

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.


Introduction
Consider two adjacent edges e 1 and e 2 in a regular polyhedron. These edges have a common vertex v and lie on the same face. We take the second face containing e 2 . In this face, there is a unique edge e 3 intersecting e 2 in a vertex different from v. Similarly, we get an edge e 4 from the edges e 2 and e 3 .
Step by step, we create a sequence e 1 , e 2 , e 3 , . . . whose elements will repeat after some times. Since we are dealing with a regular polyhedron, this is a skew polygon without selfintersections. Coxeter   In non-regular polyhedra, cyclic sequences with the property described above can have self intersections. For example, a 3-gonal bipyramid (Fig.2) has only one such sequence (up to reversing). Analogues of Petrie polygons for maps, i.e. 2-cell embeddings of graphs in closed 2-dimensional surfaces, are known as zigzags [3,5] or closed left-right paths [4,9]. General results on zigzags in plane graphs can be found in [4,Chapter 17]. A large portion of material given in [3] concerns zigzags in three-regular plane graphs related to mathematical chemistry, in particular, well-known fullerenes. In both these books, the case when there is a single zigzag is distinguished. Following [3], we call maps satisfying this condition z-knotted.
A Gauss code is a word, where each symbol occurs precisely twice [4,Section 17.4]. There is a problem related to realizing Gauss codes as closed curves with simple self-intersections in closed 2-dimensional surfaces. It is not difficult to see that a map is z-knotted if and only if there is a zigzag passing trough each edge twice. The latter is equivalent to the fact that the corresponding medial graph (the graph whose vertex set is formed by all edges of the map, and two edges are adjacent if they have a common vertex and belong to the same face, see the bold graph on Fig.3) is a realization of a certain Gauss code. We refer [4,Section 17.7] for the case of plane graphs and [1,6] for the general case. The concept of z-knottedness can be described in terms of Z 2 -homologies. First of all, we observe that every zigzag is also a zigzag in the dual map and conversely. If our map is not z-knotted, then there is a zigzag passing through some edges once. These edges form a bicycle, i.e. a cycle which is also a cycle in the dual map. By [4,Theorem 17.3.5], a plane graph is z-knotted if and only if it does not contain non-trivial bicycles. This is equivalent to the fact that the corank of the Laplacian over Z 2 is equal to 1 and well-known Kirchhoff's theorem implies that a plane graph is z-knotted if and only if the number of spanning trees in this graph is odd [4,Sections 14.15 and 17.8]. In the general case, we consider the vector space over Z 2 formed by all bicycles with the symmetric difference as an additive operation. For a z-knotted map this vector space does not contain boundaries and coboundaries and its dimension is equal to the corresponding Betti number, i.e. there is a natural one-to-one correspondence between bicycles and 1-dimensional Z 2 -homologies [1,Section 4].
In this paper, we investigate zigzags in triangulations of closed 2-dimensional surfaces. It is well-known that an n-gonal bipyramid is a z-knotted triangulation of the sphere S 2 if n is odd. Two examples of z-knotted fullerenes can be found in [3,Section 2.3] (the duals are z-knotted triangulations of S 2 ). A large class of plane z-knotted graphs (in particular, z-knotted fullerenes) was obtained using computer [3].
Our main result (Theorem 1) states that every triangulation of any closed 2dimensional surface admits a z-knotted shredding.
This statement will be proved in two steps. First, we introduce the z-monodromy M F which acts on the set of oriented edges of a face F in a triangulation. For two consecutive oriented edges e 0 and e in F we consider the unique zigzag containing the sequence e 0 , e and define M F (e) as the first (oriented) edge of F which occurs in this zigzag after e. There are precisely seven possibilities for the z-monodromies and only four possibilities can be realized in z-knotted triangulations (Theorem 2). Examples show that each of these seven possibilities happens in a certain triangulation; moreover, the four possibilities corresponding to the z-knotted case are realized in z-knotted triangulations (Section 5). If the z-monodromies of all faces in a triangulation are of z-knotted type, then this triangulation is z-knotted (Theorem 3). The second step is the gluing lemma concerning zigzags in the connected sum of triangulations (Section 6). Using this lemma, we replace every face whose z-monodromy is not of z-knotted type by some number of faces with the z-monodromies of z-knotted types and get a z-knotted shredding.
Other application of the gluing lemma is a description of all cases when the connected sum of z-knotted triangulations is z-knotted (Theorem 4).

Zigzags in triangulations
We describe some elementary properties of zigzags in triangulations (which also hold for other maps).
Let M be a connected closed 2-dimensional surface (not necessarily orientable) and let Γ be a triangulation of M , i.e. a 2-cell embedding of a connected simple finite graph in M whose faces are triangles [7, Section 3.1]. The following properties are immediately consequences of the definition: (1) every edge is contained in precisely two distinct faces, (2) the intersection of two distinct faces is an edge or a vertex or empty.
Two distinct edges are called adjacent if there is a face containing them. Since every face is a triangle, any two adjacent edges have a common vertex. Two faces are said to be adjacent if their intersection is an edge.
A zigzag in Γ is a sequence of edges {e i } i∈N satisfying the following conditions for every i ∈ N: (Z1) e i , e i+1 are adjacent, (Z2) the faces containing e i , e i+1 and e i+1 , e i+2 are distinct and the edges e i and e i+2 are disjoint.
Since Γ is finite, for every zigzag Z = {e i } i∈N there is a natural number n > 0 such that e i+n = e i for every i ∈ N. The smallest number n satisfying this condition is called the length of Z. So, our zigzag is the cyclic sequence e 1 , . . . , e n , where n is the length of Z. We say that Z is edge-simple if all edges in this cyclic sequence are mutually distinct. Note that Z can be presented as a cyclic sequence of vertices v 1 , . . . , v n , where v i and v i+1 are the vertices belonging to e i for i < n and the edge e n contains v n and v 1 . The zigzag Z is called simple if these vertices are mutually distinct. All zigzags of the Platonic solids are simple (see Fig.1). It is clear that a simple zigzag is edge-simple, but an edge-simple zigzag is not necessarily simple. Observe that every zigzag {e i } i∈N is completely determined by any pair of consecutive edges e i , e i+1 . If X = {e 1 , . . . , e n } is a sequence of edges, then X −1 denotes the reversed sequence e n , . . . , e 1 . If Z is a zigzag, then the same holds for Z −1 . If Z contains a sequence e, e ′ , then the sequence e ′ , e is contained in the reversed zigzag Z −1 . Sequences of type e, e ′ , . . . , e ′ , e are not contained in zigzags, i.e. a zigzag cannot be reversed to itself. Indeed, if such a sequence is contained in a zigzag, then this zigzag is a sequence of type e, e ′ , e 1 , e 2 , . . . , e m , e m , . . . , e 2 , e 1 , e ′ , e, . . . and there are two consecutive edges which are the same, a contradiction.
We say that Γ is z-knotted if it contains only one pair of zigzags Z, Z −1 , in other words, there is a single zigzag up to reversing. Lemma 1. If Γ is z-knotted, then each of the two zigzags passes through every edge twice. Conversely, if Γ contains a zigzag passing through every edge twice, then it is z-knotted.
This statement can be found in [3,4], but some arguments from its proof will be used in what follows. For this reason, we present this proof below.
Proof of Lemma 1. Every edge e is contained in precisely two distinct faces F 1 and F 2 . We fix a vertex on e and denote by e i , i = 1, 2 the edge in F i containing this fixed vertex and distinct from e (Fig.4). Let Z i (e), i = 1, 2 be the zigzag containing the sequence e, e i . If Γ is z-knotted, i.e. it contains only one pair of zigzags Z, Z −1 , then each of these zigzags coincides with every Z i (e) or its reverse. This implies that it passes through e twice.
If a zigzag passes through each edge twice, then it coincides with every Z i (e) or its reverse for all edges e. This is possible only in the case when Γ is z-knotted.

Main result
Let Γ and Γ ′ be triangulations of (connected closed 2-dimensional) surfaces M and M ′ , respectively. Suppose that F is a face in Γ and F ′ is a face in Γ ′ . By our assumption, F and F ′ both are homeomorphic to a closed 2-dimensional disc and each of the boundaries ∂F and ∂F ′ is the sum of three edges. Let g : ∂F → ∂F ′ be a homeomorphism transferring every vertex of F to a vertex of In what follows, such boundary homeomorphisms will be called special.
We define the connected sum Γ# g Γ ′ . We remove the interiors of F and F ′ from M and M ′ (respectively) and glue together the boundaries ∂F and ∂F ′ such that We get a triangulation of the connected sum M #M ′ which will be denoted by Γ# g Γ ′ . The vertex set of Γ# g Γ ′ is the union of the vertex sets of Γ and Γ ′ , where every v i is identified with v ′ i , and the edge set is the union of the edge sets of Γ and Γ ′ , where the edge Fig.9 and Fig.10 for non-isomorphic connected sums of 3-gonal bipyramids.
Using the connected sums of triangulations, we describe the concept of shredding of a triangulation. As above, we suppose that Γ is a triangulation of a surface M . Let F 1 , . . . , F k be mutually distinct faces of Γ and let Γ 1 , . . . , Γ k be triangulations of the sphere S 2 . For each i ∈ {1, . . . , k} we take a face F ′ i in Γ i and a special homeomorphism is a triangulation of M , where every F i is replaced by a triangulation of a 2-dimensional disc. Every triangulation of M obtained from Γ in such a way is said to be a shredding of Γ.

Z-monodromy
Let Γ be a triangulation and let F be a face in Γ whose vertices are denoted by a, b, c. Let also Ω(F ) be the set consisting of all oriented edges of F . Then where xy is the edge from x ∈ {a, b, c} to y ∈ {a, b, c}. If e is the edge xy, then we write −e for the edge yx.
Consider the permutation on the set Ω(F ) (which is the composition of two commuting 3-cycles). If x, y, z are three mutually distinct vertices of F , then D F (xy) = yz. The equality For any e ∈ Ω(F ) we take e 0 ∈ Ω(F ) such that D F (e 0 ) = e and consider the zigzag Z containing the sequence e 0 , e (recall that every zigzag is completely determined by any pair of consecutive edges). If e ′ is the first element of Ω(F ) contained in the zigzag Z after e, then we define M F (e) = e ′ , Fig.5. The transformation M F of Ω(F ) will be called the z-monodromy associated to the face F . • tetrahedrons, octahedrons and icosahedrons (three Platonic solids whose faces are triangles), • the torus triangulation obtained from a grid with diagonals, • the triangulation of the real projective plane presented on Figure 6.
In Example 4, we show that all zigzags in a (2k)-gonal bipyramid are edge-simple if k is even. Note that these zigzags are simple only for k = 2 (the octahedron case).   Proof. (1). Suppose that a zigzag passes through the edge of F containing vertices x, y ∈ {a, b, c} and goes from x to y. Then one of the following possibilities is realized: this zigzag contains the sequence xy, D F (xy), or it contains the sequence e, xy such that D F (e) = xy. In each of these cases, the zigzag belongs to Z(F ).
(2). If a zigzag belongs to Z(F ), then it contains the sequence e, e ′ , where e ∈ Ω(F ) and e ′ = D F (e). The reversed zigzag contains the sequence −e ′ , −e. Since D F (−e ′ ) = −e, it belongs to Z(F ).
(3). The set Ω(F ) consists of 6 elements, but for some distinct e, e ′ ∈ Ω(F ) the zigzags containing the sequences e, D F (e) and e ′ , D F (e ′ ) can be coincident. In this case, the reversed zigzags also are coincident.
We say that Γ is locally z-knotted for F if |Z(F )| = 2. By Lemma 2, this holds if and only if there is a single pair of zigzags Z, Z −1 containing edges of F . In the next section, we give an example for each of the seven possibilities described in Theorem 2.   Proof. If Γ is z-knotted, then it is locally z-knotted for every face F and each z-monodromy M F is one of (M1)-(M4) by Theorem 2. Now, we suppose that for every face F the z-monodromy M F is one of (M1)-(M4). It follows from Theorem 2 that Γ is locally z-knotted for all faces. Let F be a face of Γ and let Z(F ) = {Z, Z −1 }. Then Z passes through each edge of F . Therefore, if F ′ is a face adjacent to F (i.e. intersecting F in an edge), then Z belongs to Z(F ′ ) by the statement (1) from Lemma 2. Since Γ is locally z-knotted for F ′ , we have Z(F ′ ) = {Z, Z −1 }. The same holds for every face F ′ of Γ by connectedness.
Remark 1. Suppose that the z-monodromy of a face F is (M1) or (M2) and Z ∈ Z(F ). Then Z passes through each edge of F twice in the same direction, in other words, it goes through three elements of Ω(F ) twice. These elements form a cycle in D F , see Fig.7(a). In the case when the z-monodromy M F is (M3) or (M4), every zigzag from Z(F ) goes through one edge twice in the same direction and through the remaining two edges twice in opposite directions, see Fig.7(b).

Remark 2.
There is a one-to-one correspondence between cycles of the permutation D F M F and zigzags belonging to Z(F ) (see the proof of Lemma 5). An easy verification shows that |Z(F )| = 6 if M F is (M5) and we have |Z(F )| = 4 if M F is (M6) or (M7).

Examples
We describe zigzags in bipyramids and their connected sums (note that all zigzags will be presented as sequences of vertices) and give an example for each type of zmonodromy. In particular, we show that each of the z-monodromies (M1)-(M4) is realized in a z-knotted triangulation of the sphere S 2 .
which implies that M F leaves fixed e 3 and transfers e 1 to e 2 and −e 1 to −e 2 . By the statement (1) from Lemma 4, is of type (M7).

5.2.
Z-monodromy in the connected sums of bipyramids. Let BP n be as in the previous subsection. Consider another one bipyramid BP n ′ , where the vertices of the n ′ -gone are denoted by 1 ′ , . . . , n ′ and we write a ′ , b ′ for the remaining two vertices. Let S and S ′ be the faces of the bipyramids containing the vertices a, 1, 2 and a ′ , 1 ′ , 2 ′ , respectively. We describe the z-monodromies for some faces in the connected sums BP n # g BP n ′ , where g : ∂S → ∂S ′ is a special homeomorphism. One of these sums for n = n ′ = 3 is presented on Fig.9. are parts of the zigzag between two edges of the face S. Note that any two consecutive parts have the same vertex (for example, the parts A and B are joined in the vertex a). Similarly, one of the two zigzags of BP n ′ is the cyclic sequence Consider the cyclic sequence where for any two consecutive parts X, Y the last edge from X is identified with the first edge from Y . A direct verification shows that this is a zigzag in Γ. This zigzag passes through each edge of Γ twice (since it is obtained from a zigzag of BP n passing through all edges of BP n twice and a zigzag of BP n ′ satisfying the same condition). Lemma 1 implies that Γ is z-knotted. Let F be the face of Γ containing the vertices b, 1, 2 and let e 1 = 12, e 2 = 2b, e 3 = b1. Since this face appears in the zigzag as follows

and their reverses. Then
A, C ′−1 , C −1 , A ′ , B, B ′ (as in the previous example, for any two consecutive parts X, Y the last edge from X is identified with the first edge from Y ) is a zigzag in Γ. This zigzag passes through all edges of Γ twice (indeed, the sequence A, B, C contains every edge of BP n twice and A ′ , B ′ , C ′ contains every edge of BP n ′ twice). Therefore, Γ is zknotted. Let F be the face of Γ containing the vertices a, 2, 3. This face appears in the zigzag as follows . . . , a, 3, 2, . . . , 3, 2, a, . . . , 2, a, 3, . . . which implies that M F is identity.

Gluing Lemma
Let Γ and Γ ′ be triangulations. Let also F and F ′ be faces in Γ and Γ ′ , respectively. Every special homeomorphism g : ∂F → ∂F ′ induces a bijection between Ω(F ) and Ω(F ′ ) which also will be denoted by g and sends every oriented edge xy to the oriented edge g(x)g(y). The following properties of the bijection g : Ω(F ) → Ω(F ′ ) are obvious: • g(−e) = −g(e) for every e ∈ Ω(F ), A face S in a triangulation is said to be essential if every zigzag of this triangulation contains an edge from this face, or equivalently, every zigzag belongs to Z(S). It is clear that all faces in z-knotted triangulations are essential. Also, every face in a tetrahedron is essential. Lemma 6. The following assertions are fulfilled: (1) Suppose that F and F ′ are essential faces. Then the connected sum Γ# g Γ ′ is z-knotted if and only if gM F g −1 M F ′ is the composition of two distinct commuting 3-cycles. (2) Suppose that Γ ′ is z-knotted and gM F g −1 M F ′ is the composition of two distinct commuting 3-cycles. Then Γ# g Γ ′ contains a zigzag Z such that Z(S) = {Z, Z −1 } for every face S in Γ# g Γ ′ corresponding to a face of Γ ′ distinct from F ′ . Proof . . .
where m is the smallest non-zero number satisfying (gM F g −1 M F ′ ) m (e) = e.
In fact, this is a cyclic sequence X ′ 1 , X 1 , . . . , X ′ m , X m , where all X i belong to X and every X ′ i is an element of X ′ . As in the examples from the previous section, for any two consecutive parts A, B ∈ X ∪ X ′ in this sequence we identify the last edge from A with the first edge from B and get a zigzag in the connected sum Γ# g Γ ′ . We denote this zigzag by Z(e). We have X i = X −1 j and X ′ i = X ′−1 j for any pair i, j ∈ {1, . . . , m} (otherwise, the zigzag Z(e) is self-reversed which is impossible). This implies that m ≤ 3. The zigzag Z(e) corresponds to an m-cycle in the permutation gM F g −1 M F ′ and the reversed zigzag Z(e) −1 is the cyclic sequence X −1 m , X ′−1 m , . . . , X −1 1 , X ′−1 1 related to a different m-cycle in this permutation. Note that Z(e) −1 coincides with Z(e ′ ) for a certain e ′ ∈ Ω(F ′ ). As in the proof of Lemma 5, we establish that the following two conditions are equivalent: (A) the permutation gM F g −1 M F ′ is the composition of two distinct commuting 3-cycles, (B) for any e, e ′ ∈ Ω(F ′ ) the zigzag Z(e ′ ) coincides with Z(e) or Z(e) −1 .

Lemma 8.
Suppose that S is a face of Γ distinct from F . If Γ is locally z-knotted for S, then Γ# g Γ ′ also is locally z-knotted for S.