This is a descrition of the names of the columns in the excel spreadsheets Genus1501maps-chiral-data.xlsx, Genus1501maps-orientable-refl-data.xlsx and Genus1502maps-nonorientable-data.xlsx that contain information about all rotary maps of genus between 2 and 1501 (between 3 and 1502 in the case of non-orientable maps). Each row in the spreadsheet of chiral maps indicates a pair of chiral maps with the given parameters, with each memeber of the pair being the mirror image of the other. (The automorphism groups of these two maps are the same, but the presentation of each one is obtainable by replacing the generating pair (R,S) for the other by (R^-1,S^-1).)
ID: This is the name of the map. The names start with letters C, R or N, depending on whether the map is chiral, orientably regular or non-orientable. An integer representing the genus of the map is then added, which is followed by a full stop and an index that distinguishes non-isomorphic maps of the same kind and genus. The dual of a map with a name Xn.i is named Xn.i*. For example, "C11.3" is the name of the "third" chiral rotary map of genus 11 in the census and its dual has the name "C11.3*".
genus: An integer representing the genus of the map. If the , then the genus equals \(2-2\chi\), where \(\chi\) is the Euler characteristic of the surface. If the maps gas genus g, then its Euler characteristic is 2-2g (if the underlying surface of the map is orientable) or 2-g (if the surface 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.
|G|: The order of the automorphism group of the map. If the map is orientably regular or non-orientable, then |G| is 4 times the nuber of edges of the map. If it is chiral, then |G| equals twice the number of the edges.
solv: "S" or "N" depending on whether the automorphism group of the map is solvable or not.
|V|,|E|,|F|: the number of vertices, edges and faces of the maps.
mV: 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.
mF: The vertex multiplicity of the dual of the map. That is, mF is the number of edges that two adjacent faces share.
self-dual (only for orientably regular and non-orientable maps): Either "SD" or "NSD" depending on whether the map is isomorphic to its dual or not.
Wilson-inv (only for fully regular maps): A string indicating which Wilson operators the map is invariant under. There are 6 Wilson operators: I (identity), D (duality), P (Petrie duality), PD (the Petrie of the dual, sometimes called a "triality"), DP (the dual of Petrie, the inverse of PD, also called a "triality") and DPD (the dual of the Petrie of the dual, sometimes called "the opposite", and equal to PDP). These 6 operators 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 {I, P, D, DP,PD, DPD}. If X appears in this string, this means the map is invariant under the operator with the name X. For example, if the string I+P appears in this column, this means that the map is self-Petrie (invariant under the Petrie dual operator) but not under any of the other four non-trivial Wilson operators.
op-inv (only for chiral maps): This similar as the invariance under Wilson operators for reflexible maps. The difference here is that instead of the 6 Wilson operators we only have 4 operators acting on the set of chiral maps: I (identity), D (duality), M (mirror), MD (mirror of the dual, equal to the dual of the mirror). The value in this column is one of: I (meaning that the map is invariant only under the operator I), I+D (meaning that the map is self-dual but not mirror-self-dual) and I+MD (meaining that the map is mirror-self-dual but is not self-dual). Note that since the map is chiral, it cannot be invariant under M (by definition of chirality), implying that a chiral map which is invariant under D cannot be invariant under MD (and vice versa).
ExpGrp, |ExpGrp|, |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 ΔΉ koviera, 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 ExpGrp we give generators for ExpGrp(M) and in the next two columns we give its order and its index in U(q).