Totally chiral maps and hypermaps of small genus

Abstract

An orientably regular hypermap is totally chiral if it and its mirror image have no non-trivial common quotients. We classify the totally chiral hypermaps of genus up to 1001, and prove that the least genus of any totally chiral hypermap is 211, attained by twelve orientably regular hypermaps with monodromy group A 7 and type (3, 4, 4) (up to triality). The least genus of any totally chiral map is 631, attained by a chiral pair of orientably regular maps of type {11, 4}, together with their duals; their monodromy group is the Mathieu group M 11 . This is also the least genus of any totally chiral hypermap with non-simple monodromy group, in this case the perfect triple covering 3 . A 7 of A 7 . The least genus of any totally chiral map with non-simple monodromy group is 1457, attained by 48 maps with monodromy group isomorphic to the central extension 2 . Sz(8).

On chirality groups and regular coverings of regular oriented hypermaps

“…This is Theorem 1 of [5]. When M = W (H ) is Γ0-regular we can easily deduce a presentation for the monodromy group of H from a presentation of the monodromy group of M as…”

“…In [6] it was shown that (3, 3, 3) b,c is regular (as an oriented hypermap) if and only if {6, 3} b,c is regular. In [5] it was observed that (3, 3, 3) b,c is reflexible if and only if {6, 3} b,c is also reflexible. In this paper we prove that if the Walsh bipartite map [16] M = W (H ) of a regular oriented hypermap H is also regular then both M and H have the same chirality group.…”

Constructing self-dual chiral polytopes

“…The question of polytopality of P ⋄ P δ must still be addressed, but if P is a polyhedron, for example, then polytopality follows from Corollary 3.7. There are many examples of such polyhedra; for example, in [2], the authors give several examples of chiral polyhedra whose automorphism group is the Mathieu group M 11 .…”

Bicontactual Regular Hypermaps

“…For a further and deeper reading on maps we refer the reader to [1,6,13] and on hypermaps to [8,10,14]. The notation used in this paper follows that in [2,4,5].…”

“…For a further and deeper reading on maps we refer the reader to [1, 6, 13] and on hypermaps to [8,10,14]. The notation used in this paper follows that in [2,4,5].An oriented hypermap is an orientable hypermap with a fixed orientation; algebraically it is described by a triple Q = (D; R, L) consisting of a finite set of "abstract" darts D and two permutations R, L of D that generate a transitive group M on(Q) = R, L on D, called the monodromy group of Q; the orbits of R, L, and RL are the hypervertices, hyperedges, and hyperfaces, respectively. Maps are hypermaps satisfying L 2 = 1 1 and when a hypermap is not a map (L 2 = 1) it is called a proper hypermap.…”