This is a description of the names of the columns in the excel spreadsheets RegularMaps3kEData.xlsx, ChiralMaps6kEData.xlsx and LargeORMdata.xlsx. The spreadsheet RegularMaps3kEData.xlsx contains information about all fully regular maps (both orientable as well as non-orientable) with at least 2 edges and at most 3000 edges. The spreadsheet ChiralMaps6kEData.xlsx contains information about all chiral maps with at least 2 and at most 6000 edges. The spreadsheet LargeORMdata.xlsx contains information about all regular orientable maps with at least 3001 and at most 6000 edges. Below, we provide a description for the columns for each of these file separately even though there is a lot of repeatition.
ID: This is the name of the map. The name has the form RM[m;k] where m is the number of edges and k is the index of the map among all maps on m edges.
genus: The genus of the map (either orientable or non-orientable).
O/N: "O" if the map is orientable, and "N" if it is non-orientable.
p,q,r This is a triple of positive integers representing the length of each face, the valence of the underlying graph and the length of the Petrie walk in the map.
solv: "S" if the automorphism group of the map is solvable, and "N" if it is not.
|V|,|E|,|F|: the number of vertices, edges and faces of the maps.
v-mult: An integer representing the number of edges with which two adjacent vertices are connected. If the underlying graph is simple, then this is 1, otherwise it is >1. This parameter is often called the "vertex multiplicity" of the map.
f-mult: The vertex multiplicity of the dual of the map. That is, the number of edges that two adjacent faces share.
self: A string indicating which Wilson operators the map is invariant under. There are 5 non-trivial Wilson operators: Du (duality), Pe (Petrie duality), Tr (the Dual of the Petrie, sometimes called a "triality"), iT (the Petrie of the Dual, which is equal to the inverse of Tr, also sometimes called a "triality"), and Op (the Dual of the Petrie of the Dual, sometimes called "the opposite", and equal to Petrie of the Dual of the Petrie). These 5 operators, together with the identity, form a dihedral group of order 6 acting on the set of fully regular maps (orientable and non-orientable). In this column, a string of the form X+Y+Z+...
is given, where X,Y,Z ... are members of the set {Du, Pe, Op, Tr, iTs}. If X appears in this string, this means the map is invariant under the operator with the name X. For example, if the string Tr+iTr appears in this column, this means that the map is invariant under both trialities, Tr and iTr, but not under any of the other three non-trivial Wilson operators. If "none" appear in this columns, then the map is not invatiant under any of the five non-trivial Wilson operators.
Du,Pe,Op,Tr(DuPe),iT(PeDu): The names of the corresponding Wilson transforms of the map. If the map is invariant under all the Wilson transformations, then the name of the map itself will appear in all five columns.
|U(q):ExpGrp|: Given a rotary map M with valency q, and generators R and S for its rotation group G (with R representing the face-rotation and S representing the vertex-rotation, respectively), and an element j of the group U(q) of all units modulo q, one can construct a new map with the same rotation group, but now generated by the pair (R',S')=(RS^(1-j),S^j). The resulting map, denoted by H_j(M), has the same valency, but the face-length can change. The operator H_j is called the j-th hole operator, following a construction by Coxeter and developed by Wilson. Following notation introduced by Nedela and Škoviera, the subgroup of U(q) consisting of all j such that M is isomorphic to H_j(M) is called the exponent group of M, and we denote it by ExpGrp(M). In the column labelled by |U(q):ExpGrp| we give the index of ExpGrp(M) in U(q).
Hj: Here we give an information that suffice to reconstruct the H_j-transform of the map for each j in U(q). If this field is blank, then the map is invariant under all H_j (that is, |U(q):ExpGrp(M)|=1). Otherwise, a string of the form "j_1:k_1 j_2:k_2 ... j_t:k_t" appears. Here "j_1, j_2, ..., j_t" are generators of the quotient U(q)/ExpGrp(M). The number k_i is the index of the map H_{j_i}(M). For example, the stirng "7:20" in the row 323 corresponding to the map RM[30;35] means that the quotient U(q)/ExpGrp(M) is cyclic and generated by the element 7 in U(q) and that H_7(RM[30;35]) = RM[30;7].
plt: "Y" if the map is polytopal and "N" if it is not. A map is polytopal provided that no vertex of the map appears in any facial cycle more than once. A fully regular map on a group G=G<a,b,c> (where a,b,c are the generating involutions) is polytopal if and only if the intersection of the face-stabiliser <a,b> and the vertex-stabiliser <b,c> equals the edge-stabiliser <b>. For chiral maps with the group G=<R,S> (where R and S are the face-rotation and the rotation around an incident vertex, respectively), the map is polytopal if and only if the face-stabiliser <R> intersects the vertex-stabiliser <S> trivially.
plh: "Y" if the map is polyhedral and "N" if it is not. A map is polyhedral provided that its skeleton is simple and that the intersection of every pair of distinct faces in empty, a single vertex, or a single edge. Note that the skeleton of the dual of a polyhedral map is also simple. That is, the vertex- and face-multiplicity of a polyhedral map are both equal to 1. A fully regular map on a group G=G<a,b,c> (where a,b,c are the generating involutions) is polyhedral provided that the intersection of the sets <a,b><b,c> and <b,c><a,b> equals the union of the sets <a,b>, <b,c> and {a*c,b*a*c*b,a*c*b,b*c*a}. For a chiral map with the group G=<R,S> (where R and S are the face-rotation and the rotation around an incident vertex, respectively), the map is polyhedral if and only if the intersection of the sets <R><S> and <S><R> equals the union of the sets <R>, <S>, <RS> and <SR>.
Sk: The name of the skeleton of the map in the census of the skeletons of rotary maps.
PlhSk: When the map is polyhedral, the name of the skeleton of the map in the census of the skeletons of polyhedral rotary maps (otherwise blank).
ID: This is the name of the map. The name has the form CM[m;k] where m is the number of edges and k is the index of the map among all maps on m edges.
genus: The genus of the map.
p,q,r: This is a triple of positive integers representing the length of each face, the valence of the underlying graph and the length of the Petrie walk in the map.
solv: "S" if the automorphism group of the map is solvable, and "N" if it is not.
|V|,|E|,|F|: the number of vertices, edges and faces of the maps.
v-mult: An integer representing the number of edges with which two adjacent vertices are connected. If the underlying graph is simple, then this is 1, otherwise it is >1. This parameter is often called "vertex multiplicity" of the map.
f-mult: The vertex multiplicity of the dual of the map. That is, the number of edges that two adjacent faces share.
self: A string indicating which of the operators Du, Mir and DuMir the map is invariant under, where Du represents the duality, Mir represents the mirror symmetry (corresponding to keeping the embedding of the skeleton but reversing the orientation of the surface), and where DuMir represents the product of the two. Together with the trivial transformation, these three operators form the Klein 4-group acting on the set of orientable maps. Note that a map on an orientable surface is invariant under Mir if and only if it is fully regular (reflexible). In other words, a chiral map is never invariant under Mir. Hence, for a chiral map only the following three possibilities occur: it can be invariant under none of these three standard operators (then "none" appears in the corresponding field), or under Du but not under Mir and DuMir (then the field contains the string "Du"), or under DuMir (and then "DuMir" appears in the ccorresponding filed).
Du,Mir,DuMir: The names of the corresponding Wilson transforms of the map. If the map is invariant under all the Wilson transformations, then the name of the map itself will appear in all five columns.
|U(q):ExpGrp|: Same as in the file RegularMaps3kEData.xlsx. Note that since the mirror operator corresponds to the hole operator \(H_{j}\) with \(j=-1\), the index of ExpGrp(M) in U(q) is at least 2.
Hj: Same as in the file RegularMaps3kEData.xlsx.
plt,plh,Sk,PlhSk: Same as in the file RegularMaps3kEData.xlsx.
Same as in the file RegularMaps3kEData.xlsx, except that only the colums ID, genus, p, q, r, solv, |V|, |F|, v-mult, and f-mult, are given