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8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
1/66
Polycycles and
face-regular two-mapsMichel Deza
ENS/CNRS, Paris and ISM, Tokyo
Mathieu Dutour Sikiric
ENS/CNRS, Paris and Hebrew University, Jerusalem
and Mikhail Shtogrin
Steklov Institute, Moscow
p.1/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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I. Strictly
face-regular two-maps
p.2/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Definition
A strictly face-regular two-map isa
-connected
-valent map (on sphere or torus), whosefaces have size or (
-sphere or
-torus)
holds: any
-gonal face is adjacent to
-gons
holds: any -gonal face is adjacent to
-gons
-sphere
,
-sphere
,
p.3/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Euler formula
If !" $ % denote the number of edges separating - and-gon, then one has:
!
"
$
%
&
'
(
"
&
'
(
%
Euler formula)
'
0
1
2
&
3
'
3
4 with 4 being thegenus, can be rewritten as
6
'
(
"
1
6
'
(
%
&
6
3
'
3
4
This implies
!
"
$
%
7
6
'
'
1
6
'
'
8
&
!
"
$
%
9
&
@ 3
@
'
4
p.4/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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A classification
If9
A
B
, then4
&
B
, the map exists only onsphere and the number of vertices depends only on
9
.
If9
&
B
, then4
&
@
, the map exists only ontorus.
If 9
C
B
, then 4A
@
, the map exists only on
surfaces of higher genus and the number of vertices isdetermined by the genus and 9
.
Detailed classification:
On sphere:
sporadic examples + two infinite series:D
E
G I
% andP
Q E E !
R
%
On torus:
sporadic examples +@ 6
infinite cases.
p.5/4
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Some sporadic spheres
-sphere
S
,
T
-sphere
,
T
-sphere
U ,
U
@ B
-sphere
,@ B
S
p.6/4
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The parameters with sporadic tori
@ 3
-torus
S ,@ 3
V
T
-torus
S ,T
@ T
-torus
U ,@ T
V
p.7/4
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The parameters with sporadic tori
T
-torus
,T
@ B
-torus
,@ B
U
@ @
-torus
,@ @
@ 3
-torus
,@ 3
S
p.7/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -tori X , Y @ 3
They are obtained by truncating a
-valent tesselation ofthe torus by
6
-gons on the vertices from a seta
% , such
that every face is incident to exactly '6
vertices ina
% .
There is an infinity of possibilities, except for &@ 3
.
-torus
S ,
V
T
-torus
S ,T
V
b
-torus
S ,b
V
-tori
U
,
V
(
%
U
b
) are obtained (from 6above) by
-triakon (dividing
-gon into triple of
-gons) p.8/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W
c
-tori d ,c
e
Take the symbols
f g
The torus correspond to words of the form
9
S
h h h
9
p
q
with9
being equal tof
org
.
f
q
f g
q
p.9/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -tori d , r
Take the symbols
f g
The torus correspond to words of the form
9
S
h h h
9
p
q
with9
being equal tof
org
.
s
t
v
w
s
t x
v
w
p.10/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -tori y , e
If
-gons forming infinite lines, then one possibility:
Take the symbols
t x
Other tori correspond to words of the forms
v
w
with
being equal to
t
or
x
.
p.11/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -tori y , y
-gons andT
-gons are organized in infinite lines.
Only two configurations for
-gons locally:
f g
Words of the form
9
S
h h h
9
p
q
with 9 being equal to f g
or g f.
s
t x
v
w
s
t x xt
v
w
p.12/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -tori d ,
They are in one-to-one correspondence with perfectmatchings
D
of a6
-regular triangulation of the torus,such that every vertex is contained in a triangle, whose
edge, opposite to this vertex, belongs toD
.
p.13/4
i
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -tori d ,
They are in one-to-one correspondence with perfectmatchings
D
of a6
-regular triangulation of the torus,such that every vertex is contained in a triangle, whose
edge, opposite to this vertex, belongs toD
.
p.13/4
i
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -tori X ,
Given a
T
-torus, which is
andT
, the removalof edges between two
-gons produces a
-torus,which is
S and
.
Any such
-torus can be obtained in this way fromtwo
T
-tori
and
U , which are
andT
.
and
U are obtained from each other by the
transformation
p.14/4
O h
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Our research program
We investigated the cases of
-regular spheres and toribeing
or
.
Such maps with &6
should be on sphere only.
All
6
-spheres are
S .
There are infinities of
6
-spheres
for
&
B
,@
,3
; there areb
6
-spheres6
.
There are infinities of
6
-spheres
for
&
B
,@
,3
; there are two spheres
and36
spheres6
.
For a
-polyhedron, which is
, one has
.
For a
-connected
-torus, which is
, one has
6
.
p.15/4
R t ti f
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Representations of W -maps
Steinitz theorem: Any
-connected planar graph is theskeleton of a polyhedron.
Torus case:
A
-torus has a fundamental group isomorphic to
U
, its universal cover is a periodic
-plane.
A periodic
-plane is the universal cover of an
infinity of
-tori.
Take a
-torus
and its corresponding
-planeD
. If all translation preservingD
arise
from the fundamental group of
, then
is calledminimal.
Any
-plane is the universal cover of a unique
minimal torus.
p.16/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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II.
-polycycles
p.17/4
l l
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W -polycycles
A generalized
-polycycle is a3
-connected plane graphwith faces partitioned in two families
2
and2
U , so that:
all elements of2
(proper faces) are (combinatorial)
-gons;
all elements of2
U (holes, the exterior face is amongstthem) are pairwisely disjoint;
all vertices have valency
or3
and any3
-valent vertexlies on a boundary of a hole.
p.18/4
d l l s
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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W and W -polycycles
(i) Any
-polycycle is one of the following
cases:
(ii) Any
-polycycle belongs to the following
cases:
or belong to the following infinite family of
-polycycles:
This classification is very useful for classifying
-maps.
p.19/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Polycycle decomposition
A bridge is an edge going from a hole to a hole (possibly,the same).
p.20/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Polycycle decomposition
Any generalized
-polycycle is uniquely decomposablealong its bridges.
p.20/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Polycycle decomposition
The set of non-decomposable
-polycycles has beenclassified:
p.20/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Polycycle decomposition
p.20/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Polycycle decomposition
The infinite series of non-decomposable
-polycycles0
p
@
:
The only non-decomposable infinite
-polycycle are0
and0
p.20/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Polycycle decomposition
The infinite series of non-decomposable generalized
-polycyclesP
Q E E !
R
% ,
,
&
:
p.20/4
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III.
-maps
p.21/4
X - and d -cases
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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- and -cases
S
-maps exist only for
&
or
T
.For &
: infinity of spheres and minimal tori.
For &T
, the only case is strictly face-regular
T
-torus
S
,
T
.
-maps exist only for
@ B
For &
andT
: infinity of spheres and minimal tori.
For
&
b
, no sphere is known, but such tori exist:
For &@ B
, only tori exist and they are@ B
.
p.22/4
y -case
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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-case
D
E
G I
% is always
U ; so, we consider different maps.
-gons are organized in triples.
One has
@ 6
or
&
@ T
For &@
,@ 6
,@ T
, they exist only on torus and are
V
Infinity of spheres is found for
@ B
, &@ 3
and
&
@
.
Uncertain cases are &@ @
and@
. Possible sphereshave at least
b 3
and@ @ 6
vertices.
p.23/4
d - and y -cases
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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and cases
-maps are only
-tori and they are
.
U -maps exist only for &
andT
.
For &
, there is an infinity of spheres (Hajduk &Sotak) and tori.
For j
, they exist only on torus and are alsoj
k
l .
p.24/4
r -case
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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case
Possible only for6
@ 3
. The set of
-gons isdecomposed along the bridges into polycycles
0
and0
U :
For
&
@ 3
, they exist only on torus and are@ 3
S
For &@ @
, they exist only on torus and are@ @
p.25/4
r -case
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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case
For
&
, they exist only on sphere and are:
p.25/4
r -case
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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case
For
&
b
, it exist only on sphere and is:
p.25/4
r -case
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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case
For
&
T
, an infinity of
T
-spheres is known. Two toriare known, one being
T
, the other not.
For &@ B
, some spheres are known with@
B
,
B
and
b
B
vertices. Infinitness of spheres and existence oftori, which are not
@ B
U , are undecided.
p.25/4
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III. Frank-Kasper maps,
i.e. -maps
p.26/4
Frank-Kasper polyhedra
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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p p y
A Frank-Kasper polyhedron is a
6
-sphere which is6
S . Exactly
cases exists.
A space fullerene is a face-to-face tiling of the
Euclidean space0
by Frank-Kasper polyhedra. Theyappear in crystallography of alloys, bubble structures,clathrate hydrates and zeolites.
p.27/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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y y p
We consider
-spheres and tori, which are
S
The set of
-gonal faces of Frank-Kasper maps isdecomposable along the bridges into the following
non-decomposable
-polycycles:
0
m
n
o
The major skeleton
of a Frank-Kasper map is a
-valent map, whose vertex-set consists of polycycles
o
andn
. p.28/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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y y p
A Frank-Kasper
@
-sphere p.28/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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y y p
The polycycle decomposition p.28/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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E 1
C1
C3
E
1
C3
C1
C1
E1
E1
E1
E1
Their names p.28/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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E 1
C3
E
1
E1
E1
E1
E1
C1
C1
C1
The graph of polycycles. p.28/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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E 1
C3
E
1
E1
E1
E1
E1
Q
: eliminate
m
, so as to get a
-valentmap
p.28/4
Results
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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For a Frank-Kasper
-map, the gonality of faces ofthe
-valent map Q
is at mostz
%
U
{
.
If C
@ 3
, then there is no
-torus
S and there is a
finite number of
-spheres
S
.For &
@ 3
:
There is a unique
@ 3
-torus
@ 3
S
The
@ 3
-spheres@ 3
S are classified.
Conjecture: there is an infinity ofs
|
}
v
-spheres k
for
any
. p.29/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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IV.c
-maps
p.30/4
Euler formula
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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IfD
is a
-map, which is
(
-gons in isolated pairs),then:
6
'
1
3
'
1
6
'
'
@
(
%
&
on sphere
6
'
1
3
'
1
6
'
'
@
(
%
&
B
on torush
with being the number of vertices included in three
-gonal faces.
For
-maps this yields finiteness on sphere and
non-existence on torus.For
-maps this implies finiteness on sphere for
T
and non-existence on torus
p.31/4
Polycycle decomposition
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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There is no
-sphere
.
-map
, the non-decomposable
-polycycles,appearing in the decomposition are:
and the infinite serie0
U
p (see cases &@
3
below):
p.32/4
W -maps d
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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In the case
&
b
, Euler formula implies that the numberof vertices, included in three
-gons, is bounded (forsphere) or zero (for torus).
All non-decomposable
-polycycles (except thesingle
-gon) contain such vertices. This impliesfiniteness on sphere and non-existence on torus.
While finiteness of
-spheres
is proved for
&
T
and &b
, the actual work of enumeration is notfinished.
p.33/4
W
c
-toric
d and beyond
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Using Euler formula and polycycle decomposition, onecan see that the only appearing polycycles are:
@ B
-torus, which is@ B
, corresponds, in a
one-to-one fashion, to a perfect matching
D
on a6-regular triangulation of the torus, such that every
vertex is contained in a triangle, whose edge, oppositeto this vertex, belongs to
D
.
For any
@ B
, there is a
-torus, which is
.
Conjecture: there exists an infinity of
-spheres
if and only if
@ B
. p.34/4
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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V. -maps
p.35/4
Euler formula
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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The
-gons of a
U
-map are organized in rings,including triples, i.e.
-rings.
One has the Euler formula
'
'
'
(
%
1
6
'
S
1
&
on sphere
'
'
'
(
%
1
6
'
S
1
&
B
on torush
S is the number of vertices incident to
-gonalfaces and
the number of vertices incident to
-gonal faces.
It implies the finiteness for
,
6
,
.
p.36/4
All W -maps y
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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two possibilities (for
&
T
6
):
and the infinite series
p.37/4
W -maps y
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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For
&
,36
spheres and no tori. Two examples:
For
T
, there is an infinity of
-spheres andminimal
-tori, which are
U .
a
T
-map isT
U if and only if it is
U
p.38/4
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III. -maps
p.39/4
Classification for W -case
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The
-polycycles, appearing in the decomposition,are:
Consider the graph, whose vertices are -gonal faces of
a
-sphere
(same adjacency).It is a
-valent map
Its faces are3
-,
- or
-gons.
It has at most
T
vertices. p.40/4
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yields
yields
yields (one infinite series)
p.41/4
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yields
and
(two infinite series)
p.41/4
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yields
(a family
% with@
'
)
p.41/4
W -maps r
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A
-torus is
if and only if it is
.
A
-sphere, which is
, has S1
&
3 B
with
being the number of vertices contained in
-gonal
faces.For all
,
-tori,
are known:
Conj. For any
there is an infinity of
-spheres.
p.42/4
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III. -maps
p.43/4
Classification of W -maps e
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For
T
-maps, which areT
, one has
S
1
&
T
@
'
4
!
$
&
@ 3
@
'
4
with 4 being the genus (B
for sphere and@
for torus) and
the number of vertices contained in
-gonal faces.
There exists a unique
T
-torusT
:
We use for the complicated case of
-sphere
anexhaustive computer enumeration method.
p.44/4
Classification of W -maps e
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps
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Two examples amongst
T
sporadic spheres.
p.44/4
Classification of W -maps e
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One infinite series amongst
infinite series.
p.44/4
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III. -maps
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W -case
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-tori, which are
, are known for any
.For &
, they are
S .
-spheres
satisfy to ! $ &@ 3
. Is there an
infinity of such spheres?
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The smallest
-spheres
for
&
,T
,b
are:
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