a collaboration with Roman Nedela and Mária Skyvová
Here is the list of isomorphism classes of oriented edge-transitive maps on the orientable surfaces of genus \( g > 1\). The list contains all the oriented maps of small genera such that groups of orientation-preserving automorphisms have at most two orbits on the set of edges of the corresponding map. The two orbits are interchanged by an orientation-reversing automorphism of the map. Thanks to this strong conditions one can use the existing lists of discrete group actions (see e.g here) to derive full census of oriented edge-transitive maps of given genus.
The census consists of plain text files, a file for a genus. The format of record is provided with hope that records are self-explanatory. However, here is the description of data format used.
The first line of the file begins with the tex ## Begin on:
followed by the timestamp
of the creation of the file.
The line before the last line contains the number of the records in the file introduced by the text
## Checksum:
.
The first line of the file begins with the tex ## Written on:
followed by the timestamp
when file has been commited to the storage.
The file continue with the section ENUMERATION
. The maps are counted with respect to
the families defined and used in
[GW, OPPT, STW]. Counts of
non-degenerate edge-transitive maps are present. The map with simple underlying graph is
non-degenerate if every vertex, face and Petrie-walk has degree (length) at least
\(3\)
[OPPT].
The content of the file consists of the records separated by an empty line and begins immediately after the last line of the enumeration (which counts the maps of family \(5^P\)). The record consists of at most 17 text lines and contains the following data
::
. The first field holds
the identifier of the map in the form
Eg.n
, where g
is the genus \(g\) of the edge-transitive map
\(\mathbf{M}\)
and n
is the unique index in the file. This tag uniquely determines the isomorphism
class
with respect to \(\operatorname{Aut}\,\mathbf{M}\)
of edge-transitive maps in the census. Note that the map and its mirror image belong to the same
isomorphism class. The second field in the line contains the identifier of the family of
edge-transitive
maps defined in [GW, OPPT, STW].
The last field serves just for debug purposes; it keeps the information about the oriented
family
of the map as introduced in [KS]. Roughly, the correspondence among the
classes
(families) used in [GW, OPPT, STW]
and in our draft [KS] is given by the following tableE1b
E3b
E3*b
E3*c
E3c
E2
E1a
E5a
E4
E4*
E6
E3a
E3*a
E5b
StructureDescription
by GAP
[GAP]) and the
identifier of the group in SmallGroups library [BEB]. Note that that this
group
has action on the corresponding orientable surface of genus \(g\) and the corresponding
signature of
the orbifold \(\mathcal{O}\)
given in Line 2 following from the Riemmann-Hurwitz equation, \[
2g-2 = |\mathrm{Aut}^+\,\mathbf{M}|\left[2\gamma-2 + \sum_{i=1}^r
\left(1-\frac{1}{m_i}\right)\right],
\]
where \(\gamma\) is the genus the orbifold and \(\{m_1,m_2,\ldots,m_r\}\) is the multiset of
branch
indexes.
Simplify
[BC10] on the presentation of the computed finitely presented group in the
process.
The lists of relations belong are of one of finitely-presented groups (which depends on the
family
of ET map)
with generated as followsnon-degenerate
in the manner as introduced in [OPPT],
or if map is polytopal
, or polyhedral
[MT]. A
map is
polytopal, if the boundary cycle of every its face is a simple cycle. The map is
polyhedral,
if its face-width [MT] is greater that \(2\) (see e.g. here). Note that if map is polyhedral, then it is by
definition
non-degenerate, while polytopality
does not imply non-degeneracy. However, the list sometimes claims that a map is polyhedral
and non-degenerate. I decided not to
fix this inaccuracy, because this information is not misleading.reflexible
,
then there exists
the bijective mapping (the permutation of \(D\)) between the triples
\((D;R,L)\to(D;R^{-1},L^{-1})\). The
reflexible map may be selfdual
,
if there is a permutation \((D;R,L)\to(D;RL,L)\).chiral
. A chiral map may be
either positively-selfdual
or
negatively-selfdual
. The map is positively-selfdual if there is a
permutation
mapping \((D;R,L)\to(D;R^{-1}L,L)\). The map is negatively-selfdual if there is a
permutation mapping \((D;R,L)\to(D;RL,L)\). Moreover, if a map is reflexible and selfdual, then
both
former two
permutations exist.
+
` in the corresponding tag. The
map
can be
(orientably-) vertex-transitive (VT
), arc-transitive (AT
), or regular
(REG
).
Actually, if the group of orientation-preserving automorphisms is transitive on the edges of the
map, then
tag ET+
appears.
Note that the transitivity on the faces was not tested.
simple
and/or bipartite
.
v
), the number of edges (e
) and the number of faces (f
)
of
the particular map \(M\).
Although one can find the enumerations of edge-transitive maps for given genus in the corresponding
file, I
think that
it is convenient to distill the numbers in following table. Note that the number of non-degenerate maps
of
given genus belonging to a particular family is in parentheses after the number of all maps. Just recall
that map
\(\mathbf{M}\) with simple underlying graph is non-degenerate if
every vertex, face and Petrie-walk of \(\mathbf{M}\) has degree (length) at least \(3\) [OPPT].
If no map of
given family exists, then
we do not mention non-degenerate ones. Do you see some interesting patterns here?
Family Genus |
\(1\) | \(2\) | \(2^*\) | \(2\mathrm{ex}\) | \(2^*\mathrm{ex}\) | \(2^P\) | \(2^P\mathrm{ex}\) | \(3\) | \(4\) | \(4^*\) | \(4^P\) | \(5\) | \(5^*\) | \(5^P\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 10 (6) | 18 (8) | 18 (8) | 0 | 0 | 1 (0) | 0 | 44 (5) | 2 (1) | 2 (1) | 1 (0) | 0 | 0 | 0 |
3 | 20 (16) | 43 (23) | 43 (23) | 1 (1) | 1 (1) | 2 (0) | 0 | 104 (14) | 6 (3) | 6 (3) | 1 (0) | 1 (1) | 1 (1) | 0 |
4 | 20 (16) | 50 (30) | 50 (30) | 1 (1) | 1 (1) | 7 (5) | 0 | 132 (25) | 13 (5) | 13 (5) | 6 (2) | 1 (1) | 1 (1) | 0 |
5 | 26 (22) | 54 (28) | 54 (28) | 0 | 0 | 11 (6) | 0 | 166 (51) | 20 (9) | 20 (9) | 5 (1) | 0 | 0 | 0 |
6 | 23 (19) | 67 (44) | 67 (44) | 1 (1) | 1 (1) | 9 (8) | 0 | 213 (68) | 16 (6) | 16 (6) | 7 (2) | 1 (1) | 1 (1) | 0 |
7 | 21 (17) | 69 (48) | 69 (48) | 1 (1) | 1 (1) | 19 (12) | 3 (3) | 271 (124) | 38 (18) | 38 (18) | 10 (4) | 5 (2) | 5 (2) | 0 |
8 | 20 (16) | 58 (38) | 58 (38) | 0 | 0 | 9 (8) | 2 (2) | 212 (82) | 18 (9) | 18 (9) | 8 (2) | 5 (3) | 5 (3) | 1 (0) |
9 | 52 (48) | 135 (83) | 135 (83) | 2 (2) | 2 (2) | 39 (28) | 0 | 502 (215) | 77 (36) | 77 (36) | 26 (16) | 2 (2) | 2 (2) | 0 |
10 | 44 (40) | 131 (87) | 131 (87) | 3 (3) | 3 (3) | 26 (23) | 5 (5) | 480 (210) | 56 (27) | 56 (27) | 27 (15) | 12 (7) | 12 (7) | 0 |
11 | 24 (20) | 68 (44) | 68 (44) | 0 | 0 | 19 (12) | 9 (9) | 388 (235) | 58 (39) | 58 (39) | 15 (6) | 17 (8) | 17 (8) | 2 (1) |
12 | 19 (15) | 109 (90) | 109 (90) | 5 (5) | 5 (5) | 20 (19) | 4 (4) | 449 (206) | 46 (21) | 46 (21) | 10 (2) | 13 (9) | 13 (9) | 0 |
13 | 39 (35) | 149 (110) | 149 (110) | 0 | 0 | 76 (62) | 0 | 839 (519) | 154 (78) | 154 (78) | 72 (48) | 3 (3) | 3 (3) | 0 |
14 | 22 (18) | 82 (60) | 82 (60) | 2 (2) | 2 (2) | 15 (12) | 2 (2) | 346 (163) | 36 (19) | 36 (19) | 11 (2) | 7 (5) | 7 (5) | 4 (3) |
15 | 42 (38) | 170 (128) | 170 (128) | 3 (3) | 3 (3) | 27 (20) | 2 (2) | 798 (437) | 68 (38) | 68 (38) | 29 (14) | 13 (11) | 13 (11) | 2 (2) |
16 | 30 (26) | 154 (124) | 154 (124) | 4 (4) | 4 (4) | 30 (28) | 2 (2) | 738 (404) | 60 (26) | 60 (26) | 52 (30) | 12 (10) | 12 (10) | 0 |
17 | 67 (63) | 192 (125) | 192 (125) | 2 (2) | 2 (2) | 106 (89) | 7 (7) | 1214 (801) | 235 (127) | 235 (127) | 93 (68) | 22 (15) | 22 (15) | 2 (2) |
18 | 25 (21) | 140 (115) | 140 (115) | 1 (1) | 1 (1) | 23 (22) | 2 (2) | 673 (362) | 40 (16) | 40 (16) | 13 (2) | 4 (2) | 4 (2) | 2 (2) |
19 | 60 (56) | 220 (160) | 220 (160) | 6 (6) | 6 (6) | 75 (56) | 3 (3) | 1300 (832) | 203 (122) | 203 (122) | 98 (70) | 24 (21) | 24 (21) | 0 |
20 | 24 (20) | 116 (92) | 116 (92) | 1 (1) | 1 (1) | 30 (27) | 2 (2) | 714 (446) | 55 (24) | 55 (24) | 14 (2) | 6 (4) | 6 (4) | 2 (1) |
21 | 71 (67) | 243 (172) | 243 (172) | 5 (5) | 5 (5) | 84 (69) | 15 (15) | 1833 (1299) | 274 (185) | 274 (185) | 82 (49) | 31 (16) | 31 (16) | 6 (5) |
22 | 34 (30) | 170 (136) | 170 (136) | 1 (1) | 1 (1) | 30 (27) | 12 (12) | 1068 (702) | 85 (54) | 85 (54) | 55 (28) | 34 (22) | 34 (22) | 10 (8) |
23 | 19 (15) | 100 (81) | 100 (81) | 0 | 0 | 28 (18) | 0 | 768 (551) | 82 (54) | 82 (54) | 37 (16) | 0 | 0 | 0 |
24 | 31 (27) | 221 (190) | 221 (190) | 6 (6) | 6 (6) | 44 (43) | 0 | 1326 (832) | 90 (40) | 90 (40) | 16 (2) | 12 (12) | 12 (12) | 2 (2) |
26 | 30 (26) | 140 (110) | 240 (110) | 6 (6) | 6 (6) | 28 (24) | 1 (1) | 967 (653) | 84 (50) | 84 (50) | 39 (17) | 16 (15) | 16 (15) | 0 |
27 | 32 (28) | 217 (185) | 217 (185) | 2 (2) | 2 (2) | 37 (30) | 13 (13) | 1483 (1016) | 126 (87) | 126 (87) | 35 (14) | 35 (22) | 35 (22) | 4 (3) |
28 | 73 (69) | 320 (247) | 320 (247) | 6 (6) | 6 (6) | 83 (79) | 9 (9) | 1947 (1257) | 214 (125) | 214 (125) | 108 (71) | 33 (24) | 33 (24) | 0 |
Here is the actual census in the form of zipped archive: etran-catalog-20200120.zip.