to appear in Discrete and Computational Geometry
Hamiltonian submanifolds of regular polytopes
Felix Effenberger and Wolfgang Kühnel
Abstract: We investigate polyhedral -manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex -Hamiltonian if it contains the full -skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called super-neighborly triangulations) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of . By this example all regular cases of vertices with or, equivalently, all cases of regular -polytopes with are now decided.
2000 MSC classification: primary 52B70, secondary 05C45, 52-04, 53C42, 57Q35
Key words: Hamiltonian subcomplex, centrally-symmetric, tight, PL-taut, intersection form, pinched surface, sphere products
1. Introduction and results
The idea of a Hamiltonian circuit in a graph can be generalized to
higher-dimensional complexes as follows: A subcomplex of a
polyhedral complex is called -Hamiltonian
Moreover, the induced piecewise linear embedding of the surface into Euclidean -space then has the two-piece property, and it is tight [21, 2D].
Centrally-symmetric analogues can be regarded as 1-Hamiltonian subcomplexes of cross polytopes or other centrally symmetric polytopes, see . Similarly we have the genus of the -dimensional cross polytope  which is (in the orientable regular cases )
There are famous examples of quadrangulations of surfaces originally due to H. S. M. Coxeter which can be regarded as 1-Hamiltonian subcomplexes of higher-dimensional cubes , [21, 2.12]. Accordingly one talks about the genus of the -cube (or rather its edge graph) which is (in the orientable case)
see , . However, in general the genus of a 1-Hamiltonian surface in a convex -polytope is not uniquely determined, as pointed out in [39, 40]. This uniqueness seems to hold especially for regular polytopes where the regularity allows a computation of the genus by a simple counting argument.
In the cubical case there are higher-dimensional generalizations by Danzer’s construction of a power complex for a given simplicial complex . In particular there are many examples of -Hamiltonian -manifolds as subcomplexes of higher-dimensional cubes, see . For obtaining them one just has to start with a neighborly simplicial -sphere . A large number of the associated complexes are topologically connected sums of copies of . This seems to be the standard case.
Concerning triangulations of manifolds, a -dimensional simplicial complex is called a combinatorial -manifold if the union of its simplices is homeomorphic to a -manifold and if the link of each -simplex is a combinatorial -sphere. In what follows all triangulations of manifolds are assumed to be combinatorial. There exist triangulations of manifolds which are not combinatorial, for an example based on the Edwards sphere see .
With respect to the simplex as the ambient polytope a -Hamiltonian subcomplex is also called a -neighborly triangulation since any vertices are common neighbors in a -dimensional simplex. The crucial case is the case of -neighborly triangulations of -manifolds. This case was studied by the second author in . One could call this the case of super-neighborly triangulations in analogy with neighborly polytopes: The boundary complex of a -polytope can be at most -neighborly unless it is a simplex. However, combinatorial -manifolds can go beyond -neighborliness, depending on the topology. Except for the trivial case of the boundary of a simplex itself there are only a finite number of known examples of super-neighborly triangulations, reviewed in . They are necessarily tight [21, Ch.4], compare Section 5 below. The most significant ones are the unique 9-vertex triangulation of the complex projective plane , , a 16-vertex triangulation of a K3 surface  and several 15-vertex triangulations of an 8-manifold “like the quaternionic projective plane” . There is also an asymmetric 13-vertex triangulation of , but most of the examples are highly symmetric. For any -vertex triangulation of a -manifold the generalized Heawood inequality
was conjectured in ,  and later almost completely proved by I. Novik in  and proved in . Equality holds precisely in the case of super-neighborly triangulations. These are -Hamiltonian in the -dimensional simplex. In the case of 4-manifolds (i.e., ) an elementary proof was already contained in [21, 4B].
In the case of 2-Hamiltonian subcomplexes of cross polytopes the first non-trivial example was constructed by E. Sparla as a centrally-symmetric 12-vertex triangulation of as a subcomplex of the boundary of the 6-dimensional cross polytope , . Sparla also proved the following analogous Heawood inequality for the case of 2-Hamiltonian 4-manifolds in centrally symmetric -polytopes
and the opposite inequality for centrally-symmetric triangulations with vertices.
Higher-dimensional examples were found by F. H. Lutz : There are centrally-symmetric 16-vertex triangulations of and 20-vertex triangulations of . The 2-dimensional example in this series is the well known unique centrally-symmetric 8-vertex torus [22, 3.1]. All these are tightly embedded into the ambient Euclidean space . The generalized Heawood inequality for centrally symmetric -vertex triangulations of -manifolds
In the present paper we show that Sparla’s inequality for 2-Hamiltonian 4-manifolds in the skeletons of -dimensional cross polytopes is sharp for . More precisely, we show that each of the regular cases (that is, the cases of equality) for really occurs. Since the cases and are not regular, the crucial point is the existence of an example for and, necessarily, . In addition we examine the case of -Hamiltonian surfaces in the three sporadic regular 4-polytopes, see Section 2. It seems that so far no decision about existence or non-existence could be made, compare .
All cases of -Hamiltonian surfaces in the regular polytopes are decided. In particular there are no -Hamiltonian surfaces in the -cell, -cell or -cell.
All cases of -Hamiltonian -manifolds in the regular -polytopes are decided up to dimension . In particular, there is a new example of a -Hamiltonian -manifold in the boundary complex of the -dimensional cross polytope.
This follows from certain known results and a combination of Propositions 1, 2, 3, and Theorem 2 below.
The regular cases of -Hamiltonian surfaces are the following, and each case occurs:
The regular cases of -Hamiltonian -manifolds for are the following:
Here each of these cases occurs except for the case of the -simplex . Furthermore -Hamiltonian -manifolds in the -cube are known to exist for any . In the case of the -simplex the next regular case is undecided, the case occurs . The next regular case of a -octahedron is the case , see Remark 2 below.
2. Hamiltonian surfaces in the -cell, -cell, -cell
There are Hamiltonian cycles in each of the Platonic solids. The numbers of distinct Hamiltonian cycles (modulo symmetries of the solid itself) are for the cases of the tetrahedron, cube, octahedron, dodecahedron, icosahedron, see [16, pp. 277 ff.]. A 1-Hamiltonian surface in the boundary complex of a Platonic solid must coincide with the boundary itself and is, therefore, not really interesting.
Hamiltonian cycles in the regular 4-polytopes are known to exist. However, it seems that 1-Hamiltonian surfaces in the 2-skeleton of any of the three sporadic regular 4-polytopes have not yet been systematically investigated. A partial attempt can be found in .
2.1 The -cell
The boundary complex of the -cell consists of 24 vertices, 96 edges, 96 triangles and 24 octahedra. Any 1-Hamiltonian surface (or pinched surface) must have 24 vertices, 96 edges and, consequently, 64 triangles, hence it has Euler characteristic . Every edge in the polytope is in three triangles. Hence we must omit exactly one of them in each case for getting a surface where every edge is in two triangles. Since the vertex figure in the polytope is a cube, each vertex figure in the surface is a Hamiltonian circuit of length 8 in the edge graph of a cube. It is well known that this circuit is uniquely determined up to symmetries of the cube. Starting with one such vertex figure, there are four missing edges in the cube which, therefore, must be in the uniquely determined other triangles of the 24-cell. In this way, one can inductively construct an example or, alternatively, verify the non-existence. If singular vertices are allowed, then the only possibility is a link which consists of two circuits of length four each. This leads to the following proposition.
There is no -Hamiltonian surface in the -skeleton of the -cell. However, there are six combinatorial types of strongly connected -Hamiltonian pinched surfaces with a number of pinch points ranging between and and with the genus ranging between and . The case of the highest genus is a surface of genus three with four pinch points. The link of each of the pinch points in any of these types is the union of two circuits of length four.
The six types and their automorphism groups are listed in Tables 1 and 2 where the labeling of the vertices of the 24-cell coincides with the standard one in polymake .
|type||# p. pts.||orbits|
|1||10||0||, , , , , , , , ,|
|2||10||0||, , , , , , , ,|
|, , ,|
|3||8||1||, , , , , , , , ,|
|, , , , , , , ,|
|, , , , , ,|
|5||6||2||, , , , , , , , ,|
|, , , , , , , ,|
|6||4||3||, , , , , ,|
Type 1 is a pinched sphere which is based on a subdivision of the boundary of the rhombidodecahedron, see Figure 1 (left). Type 4 is just a -grid square torus where each square is subdivided by an extra vertex, see Figure 1 (right). These 16 extra vertices are identified in pairs, leading to the 8 pinch points.
Figure 1: Type 1 (left) and Type 4 (right) of Hamiltonian pinched surfaces in the -cell
Because equals the Euler characteristic of the original (connected) surface minus the number of pinch points it is clear that we can have at most 10 pinch points unless the surface splits into several components. We present here in more detail Type 6 as a surface of genus three with four pinch points, see Figure 3 (produced with JavaView). Its combinatorial type is given by the following list of 64 triangles:
The pinch points are the vertices 2, 6, 19, 23 with the following links:
The four vertices 7, 9, 15, 17 are not joined to one another and not to any of the pinch points either. Therefore the eight vertex stars of cover the 64 triangles of the surface entirely and simply, compare Figure 2 where the combinatorial type is sketched. In this drawing all vertices are 8-valent except for the four pinch points in the two “ladders” on the right hand side which have to be identified in pairs.
The combinatorial automorphism group of order 16 is generated by
The elements and generate the dihedral group of order 8 whereas commutes with and . Therefore the group is isomorphic with .
2.2 The -cell and the -cell
The 600-cell has the -vector , by duality the 120-cell has the -vector . Any 1-Hamiltonian surface in the 600-cell must have 120 vertices, 720 edges and, consequently, 480 triangles (namely, two out of five), so it has Euler characteristic and genus . We obtain the same genus in the 120-cell by counting 600 vertices, 1200 edges and 480 pentagons (namely, two out of three). The same Euler characteristic would hold for a pinched surface if there is any. We remark that similarly the 4-cube admits a Hamiltonian surface of the same genus (namely, ) as the 4-dimensional cross polytope.
There is no -Hamiltonian surface in the -skeleton of the -cell. There is no pinched surface either since the vertex link of the -cell is too small for containing two disjoint circuits.
The proof is a fairly simple procedure: In each vertex link of type the Hamiltonian surface appears as a Hamiltonian circuit of length 4. This is unique, up to symmetries of the tetrahedron and of the 120-cell itself. Note that two consecutive edges determine the circuit completely. So without loss of generality we can start with such a unique vertex link of the surface. This means we start with four pentagons covering the star of one vertex. In each of the four neighboring vertices this determines two consecutive edges of the link there. It follows that these circuits are uniquely determined as well and that we can extend the beginning part of our surface, now covering the stars of five vertices. Successively this leads to a construction of such a surface. However, after a few steps it ends at a contradiction. Consequently, such a Hamiltonian surface does not exist.
There is no -Hamiltonian surface in the -skeleton of the -cell.
This proof is more involved since it uses the classification of all 17 distinct Hamiltonian circuits in the icosahedron, up to symmetries of it [16, pp. 277 ff.]. If there is such a 1-Hamiltonian surface, then the link of each vertex in it must be a Hamiltonian cycle in the vertex link of the 600-cell which is an icosahedron. We just have to see how these can fit together. Starting with one arbitrary link one can try to extend the triangulation to the neighbors. For the neighbors there are forbidden 2-faces which has a consequence for the possible types among the 17 for them. After an exhaustive computer search it turned out that there is no way to fit all vertex links together. Therefore such a surface does not exist. At this point it must be left open whether there are 1-Hamiltonian pinched surfaces in the 600-cell. The reason is that there are too many possibilities for a splitting into two, three or four cycles in the vertex link. For a systematic search one would have to classify all these possibilities first.
The GAP programs used for the algorithmic proof of Propositions 1, 2, 3 and details of the calculations are available from the first author upon request.
3. Hamiltonian submanifolds of cross polytopes
The -dimensional cross polytope (or the -octahedron) is defined as the convex hull of the points
It is a simplicial and regular polytope, and it is centrally-symmetric with diagonals, each between two antipodal points of type and Its edge graph is the complete -partite graph with two vertices in each partition, sometimes denoted by . See  for properties of regular polytopes in general. The -vector of the cross polytope satisfies the equality
Consequently, any 1-Hamiltonian 2-manifold must have the following beginning part of the -vector:
It follows that the Euler characteristic of the 2-manifold satisfies
These are the regular cases investigated in . In terms of the genus of an orientable surface this equation reads as
This remains valid for non-orientable surfaces if we assign the genus to the real projective plane. In any case can be an integer only if . The first possibilities, where all cases are actually realized by triangulations of closed orientable surfaces , are indicated in Table 3.
Table 3: Regular cases of 1-Hamiltonian 2-manifolds
Similarly, any 2-Hamiltonian 4-manifold must have the following beginning part of the -vector:
It follows that the Euler characteristic satisfies
If we introduce the “genus” of a simply connected 4-manifold as the number of copies of which are necessary to form a connected sum with Euler characteristic , then this equation reads as
These are the “regular cases”. Again the complex projective plane would have genus here. Recall that any 2-Hamiltonian 4-manifold in the boundary of a convex polytope is simply connected since the 2-skeleton is. Therefore the “genus” equals half of the second Betti number.
Moreover, there is an Upper Bound Theorem and a Lower Bound Theorem as follows:
(E. Sparla )
If a triangulation of a -manifold occurs as a -Hamiltonian subcomplex of a centrally-symmetric simplicial -polytope then the following inequality holds
Moreover, for equality is possible only if the polytope is affinely equivalent to the -dimensional cross polytope.
If there is a triangulation of a -manifold with a fixed point free involution then the number of vertices is even, i.e., , and the opposite inequality holds
Moreover, equality in this inequality implies that the manifold can be regarded as a -Hamiltonian subcomplex of the -dimensional cross polytope.
Remark. The case of equality in either of these inequalities corresponds to the “regular cases”. Sparla’s original equation is equivalent to the one above.
By analogy, any -Hamiltonian -manifold in the -dimensional cross polytope satisfies the equation
It is necessarily -connected which implies that the left hand side is half of the middle Betti number which is nothing but the “genus”. Furthermore, there is a conjectured Upper Bound Theorem and a Lower Bound Theorem generalizing Theorem 1 where the inequality has to be replaced by
is equivalent to the one above. In particular, for any one of the “regular cases” is the case of a sphere product with (or “genus” ) and . So far examples are available for , even with a vertex transitive automorphism group see , . We hope that for there will be similar examples as well, compare Section 6.
4. -Hamiltonian -manifolds in cross polytopes
In the case of 2-Hamiltonian 4-manifolds as subcomplexes of the -dimensional cross polytope we have the “regular cases” of equality Here can be an integer only if . Table 4 indicates the first possibilities:
|8||14||7||new (Thm. 2)|
|10||42||see Remark 2|
Table 4: Regular cases of 2-Hamiltonian 4-manifolds
There is a -vertex triangulation of a -manifold which can be regarded as a centrally-symmetric and -Hamiltonian subcomplex of the -dimensional cross polytope. As one of the “regular cases” it satisfies equality in Sparla’s inequalities in Theorem with the “genus” and with .
Proof. Any 2-Hamiltonian subcomplex of a convex polytope is simply connected [21, 3.8]. Therefore such an , if it exists, must be simply connected, in particular . In accordance with Sparla’s inequalities, the Euler characteristic tells us that the middle homology group is . The topological type of is then uniquely determined by the intersection form. If the intersection form is even then by Rohlin’s theorem the signature must be zero, which implies that is homeomorphic to the connected sum of copies of , see . If the intersection form is odd then is a connected sum of 14 copies of . We will show that the intersection form of our example is even.
The induced polyhedral embedding of this manifold into 8-space is tight since the intersection with any open halfspace is connected and simply connected, compare Section 5 below. No smooth tight embedding of this manifold into 8-space can exist, see . Consequently, this embedding of into 8-space is smoothable as far as the PL structure is concerned but it is not tightly smoothable.
The -vector of this example is uniquely determined already by the requirement of 16 vertices and the condition to be 2-Hamiltonian in the 8-dimensional cross polytope. In particular there are 8 missing edges corresponding to the 8 diagonals of the cross polytope which are pairwise disjoint.
Assuming a vertex-transitive automorphism group, the example was found by using the software of F. H. Lutz described in . The combinatorial automorphism group of our example is of order 128. With this particular automorphism group the example is unique. The special element
acts on without fixed points. It interchanges the endpoints of each diagonal and, therefore, can be regarded as the antipodal mapping sending each vertex of the 8-dimensional cross polytope to its antipodal vertex in such a way that it is compatible with the subcomplex . A normal subgroup isomorphic to acts simply transitively on the 16 vertices. The isotropy group fixing one vertex (and, simultaneously, its antipodal vertex) is isomorphic to the dihedral group of order 8. The group itself is a semidirect product between and . In more detail the example is given by the three -orbits of the 4-simplices
with altogether simplices, each given by a 5-tuple of vertices out of . The group of order 128 is generated by the three permutations
The complete list of all 224 top-dimensional simplices is the following: