Hexagonal Extensions of Toroidal Maps and Hypermaps

Abstract

The rank 3 concept of a hypermap has recently been generalized to a higher rank structure in which hypermaps can be seen as "hyperfaces" but very few examples can be found in literature. We study finite rank 4 structures obtained by hexagonal extensions of toroidal hypermaps. Many new examples are produced that are regular or chiral, even when the extensions are polytopal. We also construct a new infinite family of finite nonlinear hexagonal extensions of the tetrahedron.

Highly symmetric hypertopes

“…Table 2 lists the hypertopes obtained using this construction. More precisely, they are obtained by using the following presentation for the The case where p = 6 is considerably more complicated and will be dealt with in another paper [18].…”

“…Regular hypertopes of this type are thus also classified and our conjecture extends to chiral hypertopes as well. The case where p = 6 is considerably more complicated and will be dealt with in another paper [18].…”

Faithful permutation representations of toroidal regular maps

“…The classification of regular hypertopes of ranks n − 1 and n − 2 for the symmetric group was also accomplished using a generalization of CPR graphs [6] considering Coxeter groups with nonlinear diagrams. In [10] hypertopes of rank 4, whose rank 3 residues are toroidal hypermaps, called hexagonal extensions of toroidal hypermaps, were constructed. In this work a CPR graph for the toroidal map {6, 3} (s,0) (s ≥ 2) was given.…”

Locally spherical hypertopes from generalised cubes

“…The concept was introduced in [9] with particular emphasis on regular hypertopes (that is, the ones with highest degree of symmetry). Although in [8,10,11] a number of interesting examples of regular hypertopes have been constructed, within the theory of abstract regular polytopes much more work has been done. Notably, [26] and [28] deal with universal constructions of polytopes, while in [5,23,24] some constructions with prescribed combinatorial conditions are explored.…”