A CENSUS OF ROTARY MAPS AND THEIR SKELETON GRAPHS
(by Primož Potočnik, May 2014)
This is a text file explaining the Census of all rotary maps with at least 2 and at most 3000 edges, constructed by Primož Potočnik, in May 2014. The computations were based on the census of (2,*)-groups obtained by Primož Potočnik, Pablo Spiga and Gabriel Verret (see http://www.fmf.uni-lj.si/~potocnik/work.htm).
The census is known to be complete in the range up to 1500 edges. The census is also complete up to 3000 edges for maps on oriented surfaces (both chiral and reflexible) as well as for those rotary maps on non-orientable surfaces for which at least one the Wilson tranforms lies on an orientable surface.
The description of the maps is algebraic and follows closely a previously known census of Marston Conder (see https://www.math.auckland.ac.nz/~conder/RotaryMapsWithUpTo1000Edges.txt) that contained the maps with at most 1000 edges.
Recall that a map is called rotary provided that it admits a one-step-rotation rotation around each of its vertices as well as around the centre of each face. A rotary map on an orientable surface that admits an orientation-reversing automorphism is called reflexible, while those that do not admit such an automorphism are chiral. Reflexible maps, as well as the rotary maps on non-orientable surfaces are also called regular.
REGULAR MAPS
The census of regular maps can be loaded in magma by loading the file
"ImportRegularMaps.mgm"
(requires files: "RegularMaps.txt" and "RegularMapsPlaces.txt"),
where two functions are defined:
NumberOfRegularMaps(n) ... returning the number of (non-isomorphic) regular maps with n edges, and
RegularMap(n,k) ... returning the k-th regular maps with n edges.
Each map is returned as a group G generated by three generators, a=G.1, b=G.2 and c=G.3, satisfying the relations a^2=b^2=c^2=(ac)^2=1. The orders p, q and r of ab, bc, an abc represent the face-length, the valence and the Petrie-length of the map, respectively. A regular map (G,a,b,c) is orientable provided that the subgroup is not the full group G. The number F of faces, V of vertices and E of edges equals
|G|/2p, |G|/2q and |G|/4, respectively. The genus g of underlying surface can thus be computed by the formula:
g = (2-F+E-V)/2, for orientable maps;
g = 2-F+E-V, for non-orientable maps.
By replacing the generators a and c by any pair of distinct non-trivial elements of the Klein group , one obtains a new map, called a Wilson mate of the original map. Since there are six pairs of distinct non-trivial elements, there are six Wilson mates for each map, obtained by six Wilson operators:
* The identity Id, which fixes a and c;
* Duality Du interchanges a with c and fixes ac;
* Petrie duality Pe interchanges a with ac and fixes c;
* The opposite transform interchanges c with ac and fixes a;
* The triality operator Tr induces the 3-cycle (a, ac, c) on {a,c,ac};
* The inverse triality operator IT induces the 3-cycle (a, c, ac) on {a,c,ac}.
Some precomputed data about the regular maps in the census can be found in the file
"RegularMaps.csv"
The first four columns ("Name", "#E", "orientable?", "genus") are self-explanatory. These are followed by "p", "q" and "r", corresponding to the face-length, valence and the length of the Petrie walk in the map. The column named "solv" can have value "solv", "non-solv" or "simple" corresponding to whether the group of automorphisms G is solvable, non-solvable or simple. The colums V-faith, F-faith and E-faith can have value TRUE or FALSE, depending on whether G acts faithfully on the vertices, faces or edges of the map, respectively. The next six columns are about the Wilson operators of the maps. In the column "self", the list of Wilson operators under which the map is invariant is given, while the next six columns indicate to which map is a given Wilson transform isomorphic to. The last column, named "Skeleton" gives the code of the skeleton graph of the map, as given in the census of skeletons of rotary maps (see below). The skeletons are only computed for vertex-faithful maps.
CHIRAL MAPS
The census of chiral maps can be loaded in magma by loading the file
"ImportChiralMaps.mgm",
(requires files: "ChiralMaps.txt" and "ChiralMapsPlaces.txt"),
where two functions are defined:
NumberOfChiralMaps(n) ... returning the number of (non-isomorphic) chiral maps with n edges, and
ChiralMap(n,k) ... returning the k-th regular maps with n edges.
Each map is returned as a group G generated by two generators, R=G.1 and S=G.2, satisfying the relations (RS)^2=1. The orders p and q represent the face-length and the valence of the map, respectively. The number F of faces, V of vertices and E of edges equals
|G|/p, |G|/q and |G|/2, respectively. The genus g of underlying surface can thus be computed by the formula:
g = (2-F+E-V)/2.
There are three standard transforms that can be applied to chiral maps:
* The dual can be obtained by substituting the pair (R,S) by (S,R);
* The mirror image can be obtained by taking (R,S) to (R^{-1},S^{-1});
* The mirror-dual can be obtained by taking (R,S) to (S^{-1},R^{-1}).
Some precomputed data about the chiral maps in the census can be found in the file
"ChiralMaps.csv"
The columns have a similar meaning as in the case of regular maps.
SKELETON GRAPHS
There is a separate magma and csv file containing the information about the skeleton graphs of the rotary maps. This census contains only the skeletons of vertex-faithful maps in the range up to 1500 edges. Note that if the map is not vertex-faithful, then the skeleton is not simple (it must contain parallel edges). On the other hand, even skeletons of some vertex-faithful maps are not simple. Since skeletons of rotary maps are edge-transitive, edges of such graphs come in groups of m parallel edges, where m is constant throughout the graph and is called the edge-multiplicity of the graph. The edge-multiplicity of simple graph is 1 by default. If, in a non-simple skeleton, each class of parallel edges is substituted by one edge only, we obtain the "simplification" of the skeleton. In the magma file, only simplifications of the skeletons are stored, together with the information about the edge-multiplicity.
The census of skeleton can be loaded in magma by loading the file
"Skeletons.mgm"
(requires files: "Skeletons.txt" and "SkeletonsPlaces.txt"),
Upon loading, three functions become available:
NumberOfSkeletons(n) ... returning the number of the stored skeletons on n vertices,
SkeletonGraph(n,k) ... returning the (siplification of the) k-th skeleton with n vertices,
SkeletonGraphEMult(n,k) ... returning the edge-multipilcity of the k-th skeleton on n vertices.
Some precomputed data is available in the csv file
"Skeletons.csv"
The first four columns of this spreadsheet are self-explanatory (the name, the edge-multiplicity, the order and the valence of the skeleton). The column "rotary embeddings" contains information about the rotary maps whose skeletons are isomorphic to the graph in question, and the last three columns contain the sets of the genera of these maps (separately for orientable regular maps, non-orientable regular maps and chiral maps).