Irreductible Triangulations of the Once-Punctured Torus

IRREDUCIBLE TRIANGULATIONS OF THE ONCE-PUNCTURED TORUS S. LAWRENCENKO, T. SULANKE, M.T. VILLAR, L.V. ZGONNIK, M.J. CH´ AVEZ, J.R. PORTILLO Abstract. A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the once-punctured torus, consisting of exactly 297 non-isomorphic triangulations. Keywords: triangulation of 2-manifold, irreducible triangulation, 2-manifold with boundary, punctured torus. 1. Introduction Let S∈ {Sh, Nk}be a closed surface—that is, the closed orientable (connected compact) 2-manifold Shof genus hor the closed nonorientable 2-manifold Nkof nonorientable genus k. Using this terminology, S0is the sphere, S1is the torus, N1 is the projective plane and N2is the Klein bottle. Let Dbe an open disk in S, with boundary ∂D =∂(S−D)homeomorphic to a circle. In particular, S0−D is a disk, N1−Dis the M¨obius band, and S1−Dis the punctured torus. We use the notation “Σ” whenever we assume the general case in which Σis meant to be either Sor S−D. If a graph Gis 2-cell embedded in Σ, the components of Σ−Gare called faces. Atriangulation of Σwith a simple graph G(where “simple” means “without loops Lawrencenko, S., Sulanke, T., Villar, M.T., Zgonnik, L.V., Ch´ avez, M.J., Portillo, J.R., Irreducible triangulations of the once-punctured torus. c 2018 Lawrencenko S., Sulanke T., Villar M.T., Zgonnik L.V., Ch´ avez M.J., Portillo J.R.. This work has been partially supported by P.A.I. Groups FQM-189, FQM-164 and MTM 2010-20445 (Government of Spain). and without parallel edges”) is a 2-cell embedding T:G→Σin which each face is bounded by a 3-cycle, that is, a cycle of length 3made up of 3vertices connected by 3edges of G; moreover, we demand that the closures of any two faces are either disjoint, share a single vertex, or share a single edge. We denote by V=V(T), E=E(T), and F=F(T)the sets of vertices, edges, and faces of T, respectively. The cardinality |V(T)|is called the order of T. By G(T)we denote the graph (V(T), E(T)) of T. Two triangulations T1and T2are called isomorphic (denoted T1∼ =T2) when there exists a bijection, called an isomorphism,ϕ:V(T1)→V(T2), such that [u, v, w]∈F(T1)if and only if [ϕ(u), ϕ(v), ϕ(w)] ∈F(T2). Throughout this paper we distinguish between triangulations only up to isomorphism. For Σ = S−D, let ∂T (=∂D) denote the boundary cycle of T. The vertices and edges of ∂T are called boundary vertices and boundary edges of T. A triangulation of a 2-manifold with fixed topological type is viewed as a member of the category of simplicial complexes. A triangulation is called irreducible if no edge can be contracted (to a vertex) without vacating the category of simplicial complexes or changing the topology of the underlying 2-manifold. Obstacles for edge contraction are studied in Section 3; one typical obstacle is the creation of parallel edges (forbidden in a simplicial complex). The term “irreducible triangulation” is more accurately introduced in Section 3. For the sake of brevity we abbreviate “irreducible triangulation” as “IT”, and “irreducible triangulation of the (once-) punctured torus” as “ITPT”. u v wu v0 v00 w split uw x v uw x v00 v0 edge crack uw v xy uw v00 xy v0 face crack Figure 1. Splitting (top), edge cracking (middle), face cracking (bottom). The collection of all ITs of Σis a basis for the family of all triangulations of Σ, in the sense that any triangulation of Σcan be obtained from a member of the basis by repeating the splitting operation introduced in Section 3 and illustrated by Figure 1 (top) along with two special cases in the middle and bottom. Barnette and Edelson [2] and independently Negami [23] proved that for every closed 2-manifold Sthe basis of ITs is finite. At present, bases of ITs are known for seven closed 2-manifolds: the sphere (Steinitz [24], Bowen and Fisk [4]), projective plane (Barnette [1]), torus (Lawrencenko [11, 12]), Klein bottle (Lawrencenko and Negami [20], and Sulanke [25]) as well as S2,N3, and N4(Sulanke [26, 27, 28]). In Section 2, we briefly consider the case of the sphere in the historical retrospect. In 2012 the first author (Lawrencenko) proposed [15] the problem of determining all ITs of a given 2-manifold with boundary, that is, all irreducible “punctured” triangulations with given genus and number of punctures (the latter is equal to the number of boundary components). This problem is motivated by the enormous growth of the number of ITs on closed 2-manifolds as the genus grows: already 396,784 ITs on S2, and 6,297,982 ITs on N4(see [27, p. 3]). Firstly, Boulch, Colin de Verdi`ere, and Nakamoto [3] gave upper bounds on the order of an IT of the punctured 2-manifold with given genus and number of punctures; their bounds are asymptotically tight but loose for low genus. Then, three of the authors of the present paper with participation of Quintero [7, 8] produced a complete list of the 6ITs on the M¨obius band. Before the current study, a nearly comprehensive list of 293 non-isomorphic ITPTs was already obtained with the aid of a computer program in [5] and was announced in [18]. That list missed 4ITPTs and was corrected in [6]. In this paper we develop a fully comprehensive list of 297 ITPTs (Theorem 12.1) independently and without using a computer. (The terminology used in [5, 6] is slightly different from that used in this paper.) Throughout this paper we intentionally avoid the use of a computer and all work can be checked tediously by hand. 2. The sphere In this section we examine more closely the case of the sphere; it is a very instructive one. It is not very hard to prove the following theorem; this is the content of Exercise 1 [10, p. 243] and Exercise 6 [9, p. 59]. Theorem 2.1. ([24, 4]). There is only one irreducible triangulation of the sphere: the boundary of a tetrahedron. In the historical setting, Bowen and Fisk [4] were the first who brought a modern version of Theorem 2.1; in fact, [4] contains a stronger result. However, Theorem 2.1 can be derived from the considerations in the book [24, pp. 227-229] of Steinitz and Rademacher (1934), although the authors use a different terminology. In that book it is shown that (after appropriate translation and interpretation) every 3-regular polyhedron can be obtained from a tetrahedron by a succession of the face-splitting operations. This means in dual form that the tetrahedral triangulation is a unique spherical IT. The shortest proof [17] of Theorem 2.1 uses a corollary of [21, Theorem 2, p. 264]. In fact, that corollary is a characteristic property of the sphere; it distinguishes the sphere from the rest of the 2-manifolds. The corollary states that every triangulation of the sphere contains a clean vertex—that is, a vertex vwhose link is chordless; the link of vconsists of the cycle of vertices and edges surrounding v. Then vis not incident with any non-facial 3-cycle—that is, a 3-cycle that does not bound a face of the triangulation. Thus any edge incident with vcan be contracted without creating parallel edges. One can keep iterating the edge contraction process until it terminates at some triangulation with only four faces. Although all vertices are still clean in such a tetrahedral triangulation, no edge is contractible anymore, but for another reason: after having attempted to contract an edge in the tetrahedral triangulation, one gets to a doubly covered triangle, which is not a simplicial complex. 3. Preliminaries Let Tbe a triangulation of Σ. An unordered pair of distinct adjacent edges [v, u],[v, w]of Tis called a divider of T(centered )at vertex v, denoted by hu, v, wi (=hw, v, ui). The splitting of hu, v, wi, denoted sphu, v, wi, is the operation which consists of deleting hu, v, wifrom Tand closing the resulting hole with two new triangular faces, [v0, v00, u]and [v0, v00, w], where v0and v00 denote the two split vertices; see Figure 1 (top). Under this operation, the vertex vis extended to the edge [v,0v00]and the two faces incident with this edge are inserted into the triangulation. Specifically in the case in which (Σ = S−D)AND ([u, v]∈E(T)) AND (v∈V(∂T))AND (u /∈V(∂T)), the operation sphu, v]of splitting a truncated divider hu, v]produces a single new triangular face [u, v0, v00], where [v0, v00]∈ E∂(sphu, v](T)) . Under the inverse operation, which is called contracting the edge [v0, v00], this edge collapses into a single vertex v, the faces [v0, v00, u]and [v0, v00, w]collapse into single edges [v, u]and [v, w], respectively. This operation is denoted by shiv0, v00h, which comes from the word “shrinking”, a synonym for “contracting”. Therefore, shiv0, v00hsp hu, v, wi(T)=T. It should be noticed that in the case (Σ = S− D)AND ([v0, v00]∈E(∂T)), there is only one face incident with [v0, v00]and that face collapses to a single edge under shiv0, v00h. Clearly, the operation of splitting never changes the topology of Σ. We demand that the contraction operation must preserve the topology of Σas well; moreover, parallel edges must not be created in a triangulation. In the case in which an edge ε∈E(T)occurs in some non-facial 3-cycle, if we still insist on contracting ε, parallel edges would be produced, which would exclude shiεh(T)from the category of triangulations. An edge εis called contractible or a rope (or a cable) if shiεh(T)is still a triangulation of Σ; otherwise εis called noncontractible or a rod. Therefore, one can contract ropes but not rods. The subgraph of G(T)made up of all ropes is called the rope subgraph of G(T). The only constraint to edge-contractibility in a non-tetrahedral triangulation T of a closed 2-manifold Sis determined in [1, 2, 11, 12]: an edge ε∈E(T)is a rod if and only if εsatisfies the following condition: (3.1) εis in a non-facial 3-cycle of G(T). That is, one cannot contract the edges of a non-facial 3-cycle. There are, in all, three constraints to edge-contractibility in a triangulation Tof apunctured 2-manifold S−D. Two of them are determined in [3]: an edge ε∈E(T) is a rod if and only if εsatisfies either condition (3.1) or the following condition: (3.2) εis a chord of D—that is, the end vertices of εare in V(∂D)but ε /∈E(∂D). That is, one cannot contract chords. A triangulation is said to be irreducible if it has no ropes, or in other words, each edge is a rod. For instance, a single triangle is the only IT of the disk S0−Dalthough its edges don’t meet either of conditions (3.1) and (3.2). Thus, the following is yet one more constraint to edge-contractibility of ε: (3.3) εis a boundary edge in the case the boundary cycle is a 3-cycle. Although condition (3.3) is a specific case of condition (3.1) unless S=S0and is not explicitly stated in [3], this condition deserves special mention. In the remainder of this paper we assume that S6=S0. 4. The structure of irreducible punctured triangulations As an energy metaphor (with ropes thought of as high voltage cables), a vertex of a triangulation Rof a closed 2-manifold Sis called a pylonic vertex or a ∗-vertex if that vertex is incident with all ropes of R. A triangulation that has at least one rope and at least one ∗-vertex is called a pylonic triangulation or a ∗-triangulation. In other words, Ris pylonic if and only if the rope subgraph of Ris isomorphic to the complete bipartite graph K1,n for some n. Let Tbe a triangulation of S−D. Let us restore the disk Din T, add a vertex p in Dand join pto the vertices in ∂D. We thus obtain a triangulation, T∗=T∪D, of S. We call Dthe patch, call pthe central vertex of the patch, and call T∗aparent triangulation for T. It will be shown in Section 10 that there exist single-roped triangulations of S1— that is, triangulations that have only one rope and thus two ∗-vertices. However, if a ∗-triangulation Rhas at least two ropes, Rhas a unique ∗-vertex. It is to be noted that if Tis an IT of S−D, then T∗may turn out to be an IT of S, but not necessarily. Lemma 4.1. If Tis an IT of S−D, then unless T∗is irreducible T∗is pylonic. Proof. Let εbe an edge of T, necessarily a rod. Then εsatisfies either (3.1) or (3.2). If εsatisfies (3.1), the corresponding edge ε∗in T∗also satisfies (3.1). If εsatisfies (3.2), ε∗still satisfies (3.1). In either case, ε∗is a rod. Thus, all ropes of T∗(if any) are not edges of Tand therefore are incident with the central vertex of the patch.  Corollary 4.2. A triangulation Tof S−Dis irreducible if and only if Tis obtained from a parent triangulation T∗of Seither by deleting a vertex when T∗is irreducible, or by deleting a ∗-vertex pwhen T∗is pylonic. Proof. The “only if” part follows from Lemma 4.1. The “if” part is trivial: any edge that is not incident with pbut occurs in some non-facial 3-cycle through pis a chord of the link of pand therefore is still a rod after the deletion of p. It is not a trivial question as to how the rope subgraph evolves under successively performed splittings. However, the following observation is easy to see: it is hard to make a rod out of a rope. Lemma 4.3. The only situation in which a rope in a triangulation of Schanges into a rod under a single application of the splitting operation is when the splitting is equivalent to the stellar subdivision of a face having that rope as a boundary edge. Proof. In such a situation, the edge contraction inverse to the splitting changes some rod εinto a rope. Since the contraction does not change the homotopy type of the cycles in π1(S), it follows by condition (3.1) above that εoccurs in a nonfacial, null-homotopic 3-cycle, which is possible if and only if the disk bounded by that 3-cycle is stellar subdivided.  Corollary 4.4. If a triangulation of Sis neither irreducible nor pylonic, it can never become pylonic after any sequence of splittings, except the case in which the rope subgraph is a 3-cycle forming the boundary of a face. We refer to the exceptional case of Corollary 4.4 as a “∆”. It is easy to see that a∆cannot occur in a once-split IT, but may arise in a twice-split IT of S. By the way, this is a good exercise for the reader to find an example of a twice-split IT K12 →S6in which a ∆occurs. Interestingly, ∆does not occur for the Klein bottle but does occur for S2and N3, which is obtained by searching ITs with the second author’s computer program surftri [28]. The example with ∆for S2can be joined to an IT of S1to produce an example with ∆for S3, etc. So ∆does occur for Sh (h≥2) and, similarly, for Nk(k≥3). Lemma 4.5. No ∆can ever be on the torus. The proof of Lemma 4.5 is postponed until the end of Section 11 when we will have more factual material to draw upon. By Lemma 4.5, we can restate Corollary 4.4 as follows. Corollary 4.6. If a triangulation of S1is neither irreducible nor pylonic, it can never become pylonic after any sequence of splittings. 5. The torus Throughout the remainder of this paper, we only consider triangulations of S1 or S1−D. A theorem of the first author [11, 12] states that for S1there exist, in all, twentyone non-isomorphic ITs: T1,T2, . . . , T21. They are represented in each of Figures 2, 3, and 4 with their vertices numbered by 1,2, . . . , 10; each Tiis identical for the three figures, with fixed vertex numbering. For each rectangle identify each pair of opposite sides to obtain an actual triangulation of S1. An automorphism of a triangulation Tis an isomorphism of Twith itself. The set of dividers of Tas well as the sets V(T), E(T), F(T)naturally fall into disjoint orbits under the action of the automorphism group Aut(T). The groups Aut(Ti) are determined explicitly for each i= 1,2, . . . , 21 in [13] and are reproduced here in Table I in the form of generating sets. In particular, the generating set for Aut(T2) was found in [14, p. 544]. Originally, the technique used in [13] is based on a computer program, but it is a good exercise for the reader to verify without a computer that the results in Table I are correct. (Interestingly, the reader may notice from Figure 2 that for i= 6 to 17, by appropriately dissecting the quadrilaterals into triangles, each Tican be obtained from the Cartesian product of two 3-cycles quadrangularly embedded in S1; the quadrangulation itself has a flag-transitive automorphism group of order 8×32= 72; see [16, 19, 22].) Based on Table I, we have identified the vertex, face, and edge orbits in each Ti(i= 1,2, . . . , 21). Elements in the same orbit are called similar. Figure 2 shows the vertex orbits in each Ti, where two vertices are marked by the same letter provided that the vertices are similar. Analogously, Figures 3 and 4 show the face and edge orbits, respectively. The same set of letters {a, b, c, d, . . . }is used for marking Figures 2, 3, and 4 in which, for each i, the three sets V(Ti),E(Ti), and F(Ti), respectively, are marked up independently of each other. aaa a a a aaa a aaa a 1 4 5 12 231 4 5 1 3 T1 67 aaa a a a aaaa a a a a 1 4 5 12 231 4 5 1 3 a6 T2 8 7 a a a aa 1 4 5 12 6 231 4 5 1 3 8 7 T3 b c b d dd c c bb a a a a aa 1 4 5 12 6 231 4 5 1 3 8 7 T4 bb bb bb cc c a a a a 1 4 5 12 6 231 4 5 1 3 8 T5 bb bb b b b d c c b7 a a aa a a a a a a aa a a aa 1 4 5 12 8 6 231 4 7 5 9 1 3 T6 aa a a aaa a aaa a aaa a 1 4 5 12 8 6 231 4 7 5 9 1 3 T7a aa a a a aa a aa 1 4 5 12 8 6 231 4 7 5 9 1 3 T8b b b b b a a aa aa 1 4 5 12 8 6 31 4 7 5 9 1 3 T9 bc deed cb bc aa a a aa aa 1 4 5 12 8 6 231 4 7 5 9 1 3 T10 bc bc cb dd aa a aa 1 4 5 12 8 6 231 4 7 5 9 1 3 bc c b c d de e bc aa a a a 1 4 5 12 8 6 231 4 7 5 9 1 3 bb b b ab bbb bb 2 T11 T12 Figure 2. The vertex orbits (to be continued). To count the number of vertex, face, or edge orbits in a specified Ti, we just count the number of distinct letters used for marking Tiin Figures 2, 3, or 4, respectively. In what follows we implicitly use the obvious fact that if two dividers hu1, v1, w1i and hu2, v2, w2iof Tiare similar, then the triangulations sphu1, v1, w1i(Ti)and sphu2, v2, w2i(Ti)are isomorphic. However, the converse is not generally true, as can be seen in the forthcoming sections; in particular, many counterexamples can be found in Table II. a a a a 1 4 5 12 8 6 231 4 7 5 9 1 3 T13 bb bb cc c c de ed a a a a a 1 4 5 12 8 6 231 4 7 5 9 1 3 T14 bc c b dd ee e fb aa a aa 1 4 5 12 8 6 231 4 7 5 9 1 3 T16 bc cb bb cc b a a aa 1 4 5 12 8 6 2 3 1 4 7 5 9 1 3 T15 bc bc d d ee fg hi b d aa aa 1 4 5 12 8 6 231 4 7 5 9 1 3 T17 b bc c bb c c de ef a a a a 1 4 5 12 8 6 21 4 7 5 9 1 3 T18 bc c b dd c c ee ba 3 a a aa 1 4 5 12 8 6 231 4 7 5 9 1 3 T19 bb b b c c c c de d e aa a aa 1 4 12 8 6 231 4 7 5 9 1 3 T20 b bc c c d d b ee c 5 aa aa a a aa 1 4 5 12 6 231 4 7 5 1 3 8 9 b T21 bb b b b b bb 10 Figure 2. The vertex orbits (contd.) Let Tbe an arbitrary triangulation of S1. We define the spread of a divider hu, v, wiin T, denoted |u, v, w |, to be the least distance between uand win the link of v. A divider with spread kis referred to as a k-divider. We observe from Figure 2 that the largest degree of a vertex in any Tiis 8, thus the largest spread of a divider in any Tiis 4, and thus the set of all triangulations obtainable by single splitting from toroidal ITs can be written as Λ = S4 k=1 Λk,where (1) Λk= 21 [ i=1 {sphu, v, wi(Ti) |u, v, w |=k} The spread of the divider used in generating the splitting is not an invariant of the triangulation obtained because Λ3∩Λ46=∅; for example, sph2,6,9i(T6)∼ = sph1,4,5i(T9)with |2,6,9|= 3 in T6and |1,4,5|= 4 in T9. The bottom line, however, is that we generate all triangulations of S1by repeatedly applying the splitting operation to the basis triangulations Ti. Lemma 5.1. Λ1∩Λ2=∅,Λ2∩Λ3=∅, and Λ3∩Λ1=∅. Proof. For k= 1 or 2, the ∗-vertex of any ∗-triangulation Tin Λkis incident with at least two ropes and has degree 3or 4, respectively, but any ∗-triangulation in Λ3 has either only one rope, or at least two ropes with the degree of the ∗-vertex of at least 5, and the statement follows immediately.  Corollary 5.2. Any two ITPTs obtained by deleting a ∗-vertex from parent ∗- triangulations in different sets Λ1,Λ2, or Λ3, are non-isomorphic. Proof. We assume to the contrary that two such ITPTs Tand Rare isomorphic. Then any isomorphism from Tto Rtakes ∂T onto ∂R and naturally extends to an isomorphism between the corresponding toroidal ∗-triangulations T∗and R∗, which contradicts Lemma 5.1.  Table I. The automorphism groups of the toroidal ITs. TiSet of permutations generating Aut(Ti)|Aut(Ti)| T1(1 5 3 4 7 2), (1 2 3 4 5 6 7) 42 T2(3 5)(4 7), (1 6)(3 7)(4 5), (1 5 2 7 6 3 8 4) 32 T3(2 4)(3 7), (1 6)(3 7)(5 8) 4 T4(3 5)(4 7)(6 8), (1 6 8)(5 3 2) 6 T5(2 3)(4 5)(6 8), (2 5)(3 4)(6 8) 4 T6(2 3)(4 5)(6 9)(7 8), (1 6 5 2 7 8 3 4 9) 18 T7(2 8 5 3 7 4)(6 9), (4 8)(5 7)(6 9), (1 2 3)(4 6 7)(5 8 9), (1 5 4)(2 8 6)(3 9 7) 108 T8(2 5)(3 4)(6 9), (1 5)(2 9)(3 8)(6 7), (1 9 6)(2 5 7)(3 8 4) 12 T9(1 5)(2 9)(3 8)(6 7) 2 T10 (1 5)(2 9)(3 8)(6 7), (1 6)(3 8)(5 7) 4 T11 (1 7)(2 6)(3 4)(8 9) 2 T12 (1 7 8)(2 4 9)(3 6 5), (2 3)(4 5)(6 9)(7 8), (2 4)(3 5)(7 8) 12 T13 (2 3)(4 5)(6 9)(7 8) 2 T14 (1 7)(2 9)(5 6) 2 T15 The trivial automorphism group 1 T16 (2 4)(3 5)(7 8), (1 9)(2 7)(4 8) 4 T17 (2 4)(3 5)(7 8) 2 T18 (1 9)(2 8)(3 5)(6 7) 2 T19 (2 3)(4 5)(6 9)(7 8) 2 T20 (1 9)(2 8)(3 5)(6 7) 2 T21 (2 4 3 5)(6 7 8 9), (1 6 7 9)(3 5 4 10), (1 8 6 9)(2 3 4 10) 20 Thanks to Corollary 4.2, the search for ITPTs is reduced to the search for vertex orbits in toroidal ITs and the search for toroidal ∗-triangulations, both in and out of Λ. This task naturally splits into five cases depending on the origin of the parent triangulation Tin Corollary 4.2. Table II. ITPTs: Series 3 (contd.) No. Triangulation d-vector bd-sequence 26 T9−o(0,1,4,1,2,1) (4,7,5,8) 27 T10 −e∼ =T15 −r(0,1,3,3,1,1) (4,5,6,6) 28 T10 −f∼ =T20 −o(0,2,2,2,2,1) (4,7,4,7) 29 T11 −f(0,0,3,5,1,0) (5,5,6,6) 30 T11 −l∼ =T15 −&(0,1,2,4,2,0) (4,6,5,7) 31 T11 −o∼ =T13 −h(0,1,2,4,2,0) (4,6,5,6) 32 T12 −c∼ =T14 −n(0,3,0,2,4,0) (4,6,7,6) 33 T12 −d(0,3,0,2,4,0) (4,6,4,6) 34 T13 −f∼ =T14 −f(0,2,1,3,3,0) (4,6,6,7) 35 T13 −j∼ =T18 −c(0,2,2,1,4,0) (4,6,5,7) 36 T13 −m∼ =T15 −g(0,1,3,2,3,0) (5,5,6,7) 37 T13 −n(0,1,4,0,4,0) (5,5,5,5) 38 T14 −c∼ =T17 −m(0,2,2,2,2,1) (4,5,7,6) 39 T14 −g(0,2,1,3,3,0) (4,6,4,7) 40 T14 −j∼ =T16 −e(0,2,1,4,1,1) (5,6,7,6) 41 T14 −l∼ =T15 −n(0,2,2,2,2,1) (4,5,5,6) 42 T14 −o(0,3,1,1,3,1) (4,7,5,8) 43 T15 −l∼ =T17 −e(0,1,3,3,1,1) (5,6,6,7) 44 T15 −o∼ =T19 −b(0,3,0,3,2,1) (4,6,4,7) 45 T15 −p∼ =T16 −f(0,2,1,4,1,1) (4,5,6,7) 46 T15 −t(0,1,4,1,2,1) (5,7,5,8) 47 T16 −b∼ =T17 −k(0,2,2,3,0,2) (4,5,6,5) 48 T16 −h(0,2,2,3,0,2) (5,8,5,8) 49 T16 −i(0,2,0,5,2,0) (4,7,4,7) 50 T17 −j(0,2,3,1,1,2) (4,8,5,8) 51 T17 −l(0,3,1,2,1,2) (4,7,4,8) 52 T17 −o(0,1,3,2,3,0) (4,7,5,7) 53 T18 −f∼ =T19 −n(0,3,1,1,3,1) (4,5,6,7) 54 T18 −g(0,3,1,1,3,1) (4,7,6,8) 55 T18 −k(0,4,0,0,4,1) (4,7,4,7) 56 T18 −o(0,2,2,2,2,1) (6,7,6,7) 57 T19 −d∼ =T20 −f(0,3,1,2,1,2) (4,5,7,6) 58 T19 −f(0,3,1,2,1,2) (5,7,6,8) 59 T19 −o(0,3,2,0,2,2) (5,8,5,8) 60 T20 −d(0,3,2,1,0,3) (4,8,5,8) 61 T20 −j(0,4,0,2,0,3) (4,8,4,8) 62 T20 −n(0,2,4,0,0,3) (5,8,5,8) Lemma 9.1. Each edge of Ti(i= 1,2, . . . , 20) with f-index 1 is dually reversible. Lemma 9.2. Let i∈ {1,2, . . . , 20}and let ε∈E(Ti). Assume that εgives rise to two 3-dividers. Then the two 3-dividers are similar if and only if fidx(ε) = 1. (Equivalently, they are non-similar if and only if fidx(ε) = 2.) Proof. To prove the “if” part, assume fidx(ε) = 1. Then the two 3-dividers are similar by Lemma 9.1. To prove the “only if” part, assume that the two 3-dividers are similar. We have two cases to consider: (i) the two faces incident with εare similar, and (ii) the two incident faces are non-similar. In case (i), we immediately come to the desired conclusion: fidx(ε) = 1. In case (ii), denote by vand zthe two apices corresponding to ε. Clearly, any automorphism that takes the 3-divider at vonto the 3-divider at znecessarily sends the sublink of vthat contains εonto the sublink of zthat does not contain ε, which requires that both sublinks have the same length, so the apices are both necessarily 6-valent. On the other hand, by direct inspection, we have verified that for each edge of Ti(i= 1,2, . . . , 20) with f-index 2, the apices are either non-similar or non-6-valent. Thereby, we still come to the ultimate equality: fidx(ε)=1. Given a triangulation T, we define the v-index (vertex-degree index) of a base εm∈E(T)to be the number of its apices with degree at least 6, and denote it by vidx(εm). Clearly, vidx(εm)=0,1, or 2. We define the s1-invariant as follows: (2) s1=s1(T) := X m min(fidx(εm),vidx(εm)), where the sum is taken over all edge orbits in T. The general (mth) term in Eq. (2) has value 0if and only if vidx=0, has value 2if and only if fidx=vidx=2, and has value 1in all other cases. To construct such a function, we use min(fidx, vidx). By Lemma 9.2, s1(Ti)is equal to the number of non-similar 3-dividers in Ti(i= 1,2, . . . , 20) with some of them counted twice, as explained in the next paragraph. The bases εmin Eq. (2) give rise to the counted 3-dividers. In the specific case in which the degree of a vertex, vn, is precisely 6, any two edges opposite to each other in the link of vn, when taken as bases, give rise to the same 3-divider, centered at vn, regardless of whether the two edges are similar or not. Let kidx(vn)be the χ-index (link-chirality index) of a 6-valent vertex vn∈V(T), that is, the number of distinct pairs of non-similar opposite edges in the link of vn(distinct as unordered pairs of corresponding letters). Clearly, kidx(vn)∈ {0,1,2,3}. We define the s2-invariant as follows: s2=s2(T) := X n kidx(vn), where the sum is taken over all 6-valent vertex orbits in T; if there are no 6-valent vertices, s2:= 0. It is not hard to observe that s2is equal to the number of 3-dividers doubly counted in Eq. (2). We finally come to the following simple formula: Lemma 9.3. The number of non-similar 3-dividers in Ti(i= 1,2, . . . , 20) is equal to the difference s1(Ti)−s2(Ti). The following is an example of how to intelligently obtain a complete list of pairwise non-similar 3-dividers in a given triangulation without going through a tedious check. The example addresses the “hardest triangulation” T3. Example 9.4. To illustrate the proof of Lemma 9.3, which was given prior to its statement, we consider an example of its use in determining all (pairwise) nonsimilar 3-dividers in T3. Under the action of Aut(T3), the vertices of T3fall into four orbits marked by the letters a–din Figure 2, the edges fall into eight orbits marked by a–hin Figure 4, and the faces fall into five orbits a–ein Figure 3. We choose, arbitrarily, the following eight edges as representatives of the edge orbits: edge [2,6] in orbit a,[3,4] in b,[6,7] in c,[4,5] in d,[3,8] in e,[1,5] in f,[2,4] in g, [5,8] in h. We calculate the s1-invariant by writing the edge orbits in alphabetical order and checking with Figure 3—see Table III. Table III. Calculation of the s1-invariant. edge [2,6] [3,4] [6,7] [4,5] [3,8] [1,5] [2,4] [5,8] orbit a b c d e f g h f-index 2 2 2 2 2 1 1 1 v-index 2 1 2 2 1 2 0 2 min(fidx,vidx) 2 1 2 2 1 1 0 1 s1=P = 10 s1(T3)=Σmmin(fidx(εm),vidx(εm)) = min(fidx([2,6]),vidx([2,6])) + min(fidx([3,4]),vidx([3,4])) + min(fidx([6,7]),vidx([6,7])) + min(fidx([4,5]),vidx([4,5])) + min(fidx([3,8]),vidx([3,8])) + min(fidx([1,5]),vidx([1,5])) + min(fidx([2,4]),vidx([2,4])) + min(fidx([5,8]),vidx([5,8])) = min(2,2) + min(2,1) + min(2,2) + min(2,2) + min(2,1) + min(1,2) + min(1,0) + min(1,2) =2+1+2+2+1+1+0+1=10. Now we calculate the s2-invariant. There are four orbits into which the vertices of T3fall: a, b, c, and d; see Figure 2. However, only the vertices in orbits cand dhave degree 6; we pick vertices 7and 8as their representatives (respectively). The opposite pairs of edges in the link of vertex 7are marked by letters (Figure 4) as follows: {a, a},{f, f}, and {d, d}; here kidx(7) = 0 because there is no pair of different letters. The opposite pairs of edges in the link of vertex 8 are: {c, d}, {c, d}, and {b, b}; here kidx(8) = 1 because there is only one pair of different letters, {c, d}, distinct from the other pairs. Therefore, s2(T3) = X n kidx(vn) = kidx(7) + kidx(8) = 0 + 1 = 1. Thus, T3has exactly s1−s2= 10 −1 = 9 non-similar 3-dividers. In fact, we can go even further. The proposed approach allows explicitly identifying all non-similar 3-dividers in T3; see below and check with Table III. Firstly, we inspect the edges with min(fidx,vidx) = 0 which is equivalent to vidx = 0. We discard such edges because the 3-dividers that they give rise to are, in fact, 2- or 1-dividers. In the example under consideration, we discard the edge [2,4]. Secondly, we inspect the edges with min(fidx, vidx) = 1 which is equivalent to either condition (i) (fidx = 2) AND (vidx = 1), or (ii) (fidx = 1) AND (vidx = 2), or (iii) (fidx = 1) AND (vidx = 1). In condition (i), since vidx = 1, there is only one 3-divider each such edge gives rise to; in the example under consideration, we have two edges in this case: [3,4] and [3,8]; the 3-dividers that these edges give rise to are h5,8,6iand h1,4,5i, respectively (Figure 3). In condition (ii), since fidx = 1, then there is up to similarity just one 3-divider that each such edge gives rise to, by Lemma 9.2; in the example under consideration, we have two edges in this case: [1,5] and [5,8]; the 3-dividers that the edges give rise to are h2,7,4iand h7,4,3i, respectively (Figure 3). In condition (iii), there is only one 3-divider that each such edge gives rise to, by Lemma 9.2; however, in the example under consideration, we have no edges in this case. Thirdly, among the remaining edges, we inspect the ones with min(fidx,vidx) = 2 which is equivalent to condition (fidx = 2)AND (vidx = 2). By Lemma 9.2, each such edge gives rise to two non-similar 3-dividers—namely, as seen in Figure 3, edge [2,6] gives rise to h1,4,7iand h8,3,5i(the latter is similar to h8,7,5i—Table I), edge [6,7] gives rise to h2,4,5iand h3,8,2i(similar to h7,8,4i), edge [4,5] gives rise to h6,7,1iand h3,8,2i(similar to h7,8,4i); check with Figure 3. Observe that we have a duplication: the 3-divider h3,8,2iis doubly counted because on one hand it is given rise to by two non-similar edges ([6,7] and [4,5]), but on the other hand, since vertex 8is 6-valent, these opposite (in the link of 8) edges give rise to the same 3-divider—h3,8,2i. Table IV. Calculation of the number of 3-dividers. Tis1s2The #of 3-dividers T11 0 1 T22 0 2 T310 1 9 T46 1 5 T59 0 9 T63 0 3 T71 0 1 T82 0 2 T917 3 14 T10 9 1 8 T11 22 6 16 T12 4 0 4 T13 20 3 17 T14 21 3 18 T15 40 9 31 T16 12 3 9 T17 18 3 15 T18 18 0 18 T19 21 3 18 T20 18 3 15 T21 1 0 2∗ P= 217 It is easy to count s1−s2with help from Figures 3, 4. The results are collected in Table IV; this table can be regarded as a corollary of Lemma 9.3; the asterisk in the last row indicates that the formula in Lemma 9.3 does not apply for T21 addressed in Lemma 9.5. Moreover, similarly to Example 9.4, we have produced a complete list of non-similar 3-dividers in each Tifor i= 1,2, . . . , 20; there are totally 215 3-dividers. Check with Table IV. In addition, two more are found in T21: Lemma 9.5. If an edge εof T21 gives rise to at least one 3-divider, then εis similar to the edge [2,6] (Figure 4) which, in fact, gives rise to two non-similar 3-dividers: h5,10,7iand h5,4,1i(the latter is similar to h5,10,8i—Table I). Proof. All edges of T21 split into two orbits—aand b(see Figure 4). For each edge in orbit bas the base, both apices are 4-valent. Therefore, no edge in bgives rise to a 3-divider. On the other hand, each edge in orbit ais dually irreversible (check with Figure 4 and Table I) and thereby gives rise to two non-similar 3-dividers even though it has f-index 1(Figure 3).  Table V. Λ3with isomorphic duplications (to be continued). Triangulations Total sph1,7,6i(T1)1 sph7,8,5i(T2) sph6,8,1i(T2)2 sph1,4,7i(T3) sph1,4,5i(T3) sph2,4,5i(T3) sph7,4,3i(T3) sph6,7,1i(T3) sph8,7,5i(T3) sph2,7,4i(T3) sph7,8,4i(T3) sph5,8,6i(T3)9 sph4,3,2i(T4)sph4,3,6i(T4) sph8,3,2i(T4)sph6,3,1i(T4) sph8,7,5i(T4)5 sph6,2,5i(T5)sph6,2,7i(T5)sph3,2,1i(T5) sph3,2,5i(T5) sph8,2,4i(T5) sph8,2,1i(T5) sph7,2,4i(T5) sph3,7,2i(T5) sph8,7,6i(T5)9 sph8,6,3i(T6)sph7,6,4i(T6) sph9,6,2i(T6)3 sph4,6,7i(T7)1 sph1,8,9i(T8)sph2,8,4i(T8)2 sph2,9,4i(T9) sph2,9,7i(T9) sph3,9,6i(T9) sph3,9,7i(T9) sph5,9,8i(T9) sph5,9,6i(T9)sph4,9,8i(T9) sph1,8,6i(T9) sph5,8,9i(T9) sph4,8,2i(T9) sph3,4,5i(T9)sph5,4,2i(T9) sph2,4,7i(T9) sph6,4,3i(T9)14 sph5,4,2i(T10) sph5,4,3i(T10) sph8,9,5i(T10) sph2,9,4i(T10) sph7,8,5i(T10) sph1,8,7i(T10)sph9,8,4i(T10) sph9,8,5i(T10)8 sph3,6,8i(T11)sph7,6,4i(T11) sph9,6,2i(T11) sph7,3,5i(T11) sph7,3,9i(T11) sph6,3,5i(T11) sph6,3,1i(T11) sph2,3,1i(T11) sph2,3,4i(T11) sph9,3,4i(T11) sph4,5,3i(T11)sph8,5,9i(T11) sph1,5,7i(T11) sph6,8,1i(T11) sph9,8,5i(T11) sph2,8,4i(T11)16 sph9,6,4i(T12) sph4,6,7i(T12) sph8,6,3i(T12) sph5,6,3i(T12)4 sph1,2,7i(T13) sph1,2,3i(T13) sph5,2,7i(T13) sph5,2,6i(T13) sph8,2,4i(T13) sph8,2,6i(T13)sph3,2,4i(T13) sph9,4,1i(T13) sph9,4,2i(T13) sph7,4,2i(T13) sph7,4,6i(T13) sph3,4,6i(T13) sph3,4,5i(T13) sph1,4,5i(T13) sph5,6,7i(T13) sph4,6,9i(T13)sph2,6,8i(T13)17 sph5,6,3i(T14)sph5,6,7i(T14) sph4,6,7i(T14) sph4,6,9i(T14) sph2,6,8i(T14) sph2,6,9i(T14) sph3,6,8i(T14) sph2,3,5i(T14) sph2,3,4i(T14) sph8,3,1i(T14) sph1,3,6i(T14)sph3,9,7i(T14) sph8,9,4i(T14) sph6,9,5i(T14) sph9,4,1i(T14) sph9,4,2i(T14)sph3,4,5i(T14) sph1,4,5i(T14)18 sph1,2,3i(T15) sph5,2,6i(T15) sph8,2,4i(T15) sph9,4,1i(T15) sph9,4,2i(T15) sph7,4,2i(T15)sph7,4,6i(T15) sph3,4,6i(T15) sph3,4,5i(T15) sph1,4,5i(T15) sph6,8,3i(T15)sph7,8,2i(T15) sph9,8,5i(T15) sph9,3,4i(T15) sph9,3,6i(T15) sph5,3,7i(T15) sph5,3,2i(T15) sph1,3,8i(T15) sph1,3,6i(T15) sph4,3,2i(T15) sph7,3,8i(T15)sph2,6,8i(T15) sph3,6,5i(T15) sph7,6,4i(T15) sph9,5,8i(T15) sph9,5,2i(T15) sph4,5,1i(T15) sph4,5,2i(T15) sph6,5,3i(T15) sph6,5,1i(T15) sph8,5,3i(T15)31 sph5,8,9i(T16)sph6,8,3i(T16) sph7,8,2i(T16) sph4,6,7i(T16) sph3,6,5i(T16) sph2,3,5i(T16) sph2,3,4i(T16) sph8,3,1i(T16) sph1,3,6i(T16)9 sph9,3,4i(T17) sph9,3,6i(T17) sph5,3,2i(T17) sph5,3,7i(T17) sph1,3,8i(T17) sph1,3,6i(T17)sph4,3,2i(T17) sph7,3,8i(T17) sph1,2,3i(T17) sph5,2,6i(T17) sph8,2,4i(T17) sph2,6,9i(T17) sph3,6,5i(T17) sph7,6,4i(T17) sph7,6,5i(T17)15 Table V. Λ3with isomorphic duplications (contnd.) Triangulations Total sph2,8,7i(T18) sph2,8,4i(T18) sph5,8,4i(T18) sph5,8,9i(T18) sph6,8,3i(T18) sph6,8,9i(T18)sph7,8,3i(T18) sph2,3,5i(T18) sph2,3,1i(T18) sph8,3,1i(T18) sph8,3,4i(T18)sph9,3,4i(T18) sph9,3,7i(T18) sph5,3,7i(T18) sph7,4,2i(T18) sph3,4,6i(T18)sph3,4,9i(T18) sph2,4,9i(T18)18 sph5,7,3i(T19)sph6,7,4i(T19) sph2,7,8i(T19) sph4,2,8i(T19) sph4,2,3i(T19) sph1,2,3i(T19)sph1,2,7i(T19) sph5,2,6i(T19) sph5,2,7i(T19) sph8,2,6i(T19) sph1,4,8i(T19)sph1,4,5i(T19) sph2,4,7i(T19) sph2,4,9i(T19) sph6,4,3i(T19) sph6,4,8i(T19)sph5,4,7i(T19) sph9,4,3i(T19)18 sph5,8,9i(T20)sph7,8,3i(T20) sph4,8,2i(T20) sph2,3,5i(T20) sph2,3,4i(T20) sph8,3,1i(T20) sph8,3,7i(T20) sph9,3,6i(T20) sph9,3,4i(T20) sph5,3,7i(T20) sph1,3,6i(T20)sph7,4,5i(T20) sph3,4,9i(T20) sph1,4,8i(T20) sph6,4,8i(T20)15 sph5,10,7i(T21)sph5,10,8i(T21)2 Using Lemma 9.3 (as in Example 9.4) and Lemma 9.5, we collect in Table V the 217 triangulations produced by splitting all the 217 non-similar 3-dividers in Ti (i= 1,2, . . . , 21), which provides the whole set Λ3(with isomorphic duplications). 10. The search for ITPTs: Case 3: Series 4-6 In order to apply Corollary 4.2, we identify ∗-triangulations from the 217 triangulations in Table V. Moreover, we retain only non-isomorphic triangulations. It is a matter of mere inspection to obtain the following lemma. Lemma 10.1. There are precisely eleven ∗-triangulations in Λ3, as follows: sph1,7,6i(T1),sph6,8,1i(T2),sph7,4,3i(T3),sph2,7,4i(T3),sph7,8,4i(T3), sph5,8,6i(T3),sph6,3,1i(T4),sph3,2,1i(T5),sph3,2,5i(T5),sph3,7,2i(T5), and sph8,7,6i(T5). These triangulations are boxed in Table V and are also presented in Figure 5 with all ropes in bold type. It should be noticed that eight of the eleven ∗-triangulations in Lemma 10.1 have only one rope and two ∗-vertices. The removal of a ∗-vertex in each of the eleven triangulations creates the following 19 ITPTs: sph1,7,6i(T1)−70,sph1,7,6i(T1)−700,sph6,8,1i(T2)−80, sph6,8,1i(T2)−800,sph7,4,3i(T3)−40,sph7,4,3i(T3)−400, sph2,7,4i(T3)−70,sph2,7,4i(T3)−700,sph7,8,4i(T3)−80, sph5,8,6i(T3)−80,sph5,8,6i(T3)−800,sph6,3,1i(T4)−300, sph3,2,1i(T5)−20,sph3,2,5i(T5)−20,sph3,2,5i(T5)−200, sph3,7,2i(T5)−70,sph3,7,2i(T5)−700,sph8,7,6i(T5)−70, sph8,7,6i(T5)−700. However, fourteen of these are isomorphic in pairs as follows: sph1,7,6i(T1)−70∼ =sph1,7,6i(T1)−700,sph6,8,1i(T2)−80∼ =sph6,8,1i(T2)−800, sph2,7,4i(T3)−70∼ =sph2,7,4i(T3)−700,sph7,8,4i(T3)−80∼ =sph3,2,1i(T5)−20, sph5,8,6i(T3)−80∼ =sph5,8,6i(T3)−800,sph3,7,2i(T5)−70∼ =sph3,7,2i(T5)−700, sph8,7,6i(T5)−70∼ =sph8,7,6i(T5)−700. 1 4 5 12 231 4 5 1 sph1,7,6i(T1) 670 3 1 4 5 12 231 4 5 1 3 sph6,8,1i(T2) 7800 6 80 700 1231 400 6 57 8 5 40 1231 1231 46 5700 85 4 1231 40 400 sph7,4,3i(T3) sph2,7,4i(T3) 70 1231 46 575 4 1231 sph7,8,4i(T3) 800 12301 46 5785 4 12301 sph6,3,1i(T4) 12031 46 5 7 85 4 12031 sph3,2,1i(T5)sph3,2,5i(T5) sph3,7,2i(T5)sph8,7,6i(T5) 300 200 1231 46 57 80 5 4 1231 sph5,8,6i(T3) 800 80 12031 46 5 7 85 4 12031 200 1231 46 57085 4 1231 700 1231 46 57085 4 1231 700 Figure 5. Pylonic triangulations in Λ3. We thus get Series 4 amounting to 19 −14/2 = 12 non-isomorphic ITPTs: Lemma 10.2. There are precisely 12 ITPTs obtainable by deleting a ∗-vertex from a∗-triangulation in Λ3. The twelve triangulations form Series 4 and are collected in Table VI. Proof. Actually, it remains to verify that these twelve triangulations are non-isomor- phic. This follows from the fact that they have either differing d-vectors or bdsequences; check with Table VI.  Table VI. ITPTs: Series 4. No. Triangulation d-vector bd-sequence 1sph1,7,6i(T1)−70(0,1,2,4) (4,6,5,5,6) 2sph6,8,1i(T2)−80(0,1,2,5,0) (4,6,5,5,6) 3sph7,4,3i(T3)−40(0,3,0,5,0) (4,6,4,6,4,6) 4sph7,4,3i(T3)−400 (0,0,5,2,1) (5,6,5,5,6) 5sph2,7,4i(T3)−70(0,2,2,2,2) (4,7,5,4,7) 6sph7,8,4i(T3)−80(0,2,2,2,2) (4,6,4,5,7) 7sph5,8,6i(T3)−80(0,1,3,3,1) (4,5,5,6,6) 8sph6,3,1i(T4)−300 (0,2,4,0,2) (4,5,5,4,5,5) 9sph3,2,5i(T5)−20(1,2,1,2,2) (3,6,4,7,4,7) 10 sph3,2,5i(T5)−200 (0,2,3,0,3) (4,5,7,5,7) 11 sph3,7,2i(T5)−70(0,3,1,1,3) (4,6,7,4,7) 12 sph8,7,6i(T5)−70(0,2,2,2,2) (4,5,6,6,5) It is possible to produce more ∗-triangulations by further splitting the ∗-triangula- tions in Lemma 10.1 that have a unique rope, such as sph6,8,1i(T2)(Figure 5). Some of the twice-split triangulations are out of Λ, but some of them may have a rope whose contraction yields an IT; for instance, shi6, phsph6,80,1isph6,8,1i(T2)∼ =T16, where pstands for the ∗-vertex. Thus the latter twice-split triangulations remain in Λ! Lemma 10.3. Splitting a ∗-triangulation T∗in Lemma 10.1 (Figure 5) still produces a ∗-triangulation if and only if the splitting is equivalent to: (a) the stellar subdivision of either of the two faces incident with a rope provided that rope is a unique rope of T∗, or (b) the cracking of a rope provided that rope is a unique rope of T∗. Proof. The “if” part is obvious. In proving the “only if” part, observe from Figure 5 that there are two cases to consider as follows. Case “T∗has only one rope”. Observe from Figure 5 that the degrees of the end vertices of the rope are 5or 6. Furthermore, splitting any divider centered at a5or 6-valent end vertex certainly destroys the pylonicity of T∗unless it meets condition (a) or (b); the only questionable situation is if the center is 6-valent and the divider has spread 3; then an additional consideration is as follows. There are precisely two 6-valent end vertices of a rope—vertex 40in sph7,4,3i(T3)and vertex 20in sph3,2,5i(T5). By inspection, we verify that the following six triangulations produced by splitting a 3-divider are not pylonic: sph1,40,7isph7,4,3i(T3), sph2,40,400isph7,4,3i(T3), sph6,40,3isph7,4,3i(T3), sph6,20,5isph3,2,5i(T5), sph4,20,200isph3,2,5i(T5), sph1,20,3isph3,2,5i(T5). Case “T∗has precisely two ropes”. In fact, there are only three such triangulations in Figure 5: T∗∈ {sph7,8,4i(T3),sph6,3,1i(T4),sph3,2,1i(T5)}. Observe from Figure 5 that each T∗contains precisely two ropes—which we denote by [p, y]and [p, z]—and also observe that the degree of the central ∗-vertex pis 5or 6, and that |y, p, z |≥ 2. Assume that splitting sphu, v, wiof T∗produces a ∗-triangulation. Then, necessarily, v=p, since otherwise the newly produced edge [v0, v00]would not be incident with the ∗-vertex p. Furthermore, it is not hard to prove that for preserving the pylonicity property, it is necessary that hu, v, wi=hu, p, wiis a 3- divider that does not cross hy, p, ziat pand is edge disjoint from hy, p, zi. Since |y, p, z |≥ 2, such a situation is theoretically possible but requires the degree of p to be at least 7. By Corollary 4.6, there are not any other ∗-triangulations that belong to Λ3 or can be obtained from a member of Λ3by splitting. By Corollary 4.2, the corresponding ITPTs are obtained from Figure 5: (a) by the removal of either of the two faces incident with a single rope, and (b) by the removal of a single rope. In case (a), we have to inspect the sixteen triangulations obtained by the face removal from the eight single-roped triangulations in Lemma 10.1. These sixteen are naturally paired with each other in Table VII. The triangulations in each pair ]1−6are isomorphic, which is verified straightforwardly. The ones in pair ]7are not isomorphic because they have differing bd-sequences (check with Table VII). Moreover, as shown in the next paragraph, the ones in pair ]8are not isomorphic even though they have the same d-vector and the same bd-sequence. To show that the triangulations sph3,2,5i(T5)−[5,20,200]and sph3,2,5i(T5)− [3,20,200]in pair ]8of Table VII are not isomorphic, we observe on one hand that any such isomorphism would fix the single rope [20,200], swapping the apices 5and 3. On the other hand, the degrees of the neighboring vertices are ordered differently in the links of vertices 5and 3, and hence vertices 5and 3are non-similar in sph3,2,5i(T5) and hence no such isomorphism is possible. There are no more isomorphic pairs in Table VII except the above-mentioned six pairs because the rest of the pairs have differing d-vectors or bd-sequences (except pair ]8). Therefore, Table VII provides Series 5 of 10 non-isomorphic ITPTs. In case (b) we obtain eight more ITPTs from the single-roped triangulations in Figure 5 by the rope removal. However, four of them are isomorphic to the ones originated from Λ2and already present in Series 3 (Table II): sph6,8,1i(T2)−[80,800]∼ =T16 −i, sph7,4,3i(T3)−[40,400]∼ =T17 −o, sph5,8,6i(T3)−[80,800]∼ =T14 −g, sph8,7,6i(T5)−[70,700]∼ =T12 −d. The remaining four form Series 6 of ITPTs, collected in Table VIII; those four are pairwise non-isomorphic because they have differing d-vectors. Case 4. Parent triangulation T∗is in Λ4or can be obtained from a member of Λ4by a sequence of splittings. By Eq. (1), if T∗= sphu, v, wi(Ti)∈Λ4, then hu, v, wiis a 4-divider. A comprehensive list of non-similar 4-dividers can be generated by a sort of inclusion-exclusion technique like that introduced in Section 9. We omit the details here since it is not very hard to determine such a list directly, using Table I. There are, in total, 42 non-similar 4-dividers in the list. They are collected in Table IX in the form of the corresponding splittings of the corresponding ITs. The resulting split-up triangulations collectively form the whole set Λ4(with isomorphic duplications). It is a matter of a routine inspection to verify the following. Table VII. ITPTs: Series 5. No. No. of of Triangulation d-vector bd-sequence ITPT pair 1 1 sph1,7,6i(T1)−[6,70,700]∼ =(0,0,2,4,2) (5,5,7) ∼ =sph1,7,6i(T1)−[1,70,700] 2 2 sph6,8,1i(T2)−[6,80,800]∼ =(0,0,2,5,2,0) (5,5,7) ∼ =sph6,8,1i(T2)−[1,80,800] 3 3 sph7,4,3i(T3)−[7,40,400]∼ =(0,0,3,3,3,0) (5,6,7) ∼ =sph7,4,3i(T3)−[3,40,400] 4 4 sph2,7,4i(T3)−[4,70,700]∼ =(0,0,4,3,0,2) (5,5,8) ∼ =sph2,7,4i(T3)−[2,70,700] 5 5 sph3,7,2i(T5)−[3,70,700]∼ =(0,1,4,0,2,2) (5,5,8) ∼ =sph3,7,2i(T5)−[2,70,700] 6 6 sph8,7,6i(T5)−[6,70,700]∼ =(0,1,2,2,4,0) (5,5,6) ∼ =sph8,7,6i(T5)−[8,70,700] 7 7 sph5,8,6i(T3)−[6,80,800](0,0,3,3,3,0) (5,5,6) 8 7 sph5,8,6i(T3)−[5,80,800](0,0,3,3,3,0) (5,5,7) 9 8 sph3,2,5i(T5)−[5,20,200](0,1,3,2,1,2) (5,8,6) 10 8 sph3,2,5i(T5)−[3,20,200](0,1,3,2,1,2) (5,8,6) Table VIII. ITPTs: Series 6. No. Triangulation d-vector bd-sequence 1sph1,7,6i(T1)−[70,700](0,2,0,4,2) (7,4,7,4) 2sph2,7,4i(T3)−[70,700](0,2,2,3,0,2) (4,8,4,8) 3sph3,2,5i(T5)−[20,200](0,2,3,1,1,2) (4,8,5,8) 4sph3,7,2i(T5)−[70,700](0,3,2,0,2,2) (4,8,4,8) 11. The search for ITPTs: Case 4 Lemma 11.1. None of the triangulations in Λ4are pylonic. Thus, by Lemma 11.1 and Corollaries 4.2, 4.6, there are no ITPTs which can be derived in Case 4. Proof of Lemma 4.5: It is not hard to prove that a ∆may occur only after the second consecutive splitting of an IT and only if the first applied splitting produces a single rope, ε. We know from the above that a single rope may only occur as described above in the proof of Lemma 10.3 (case “T∗has only one rope”), in