Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps

8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps

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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


2/66

I. Strictly

face-regular two-maps

p.2/4


3/66

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


4/66

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


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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


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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


10/66

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


11/66

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


12/66

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


13/66

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


14/66

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


15/66

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


16/66

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


17/66

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


18/66

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


19/66

II.

-polycycles

p.17/4

l l


20/66

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


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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


22/66


A bridge is an edge going from a hole to a hole (possibly,the same).

p.20/4



23/66


Any generalized

-polycycle is uniquely decomposablealong its bridges.

p.20/4



24/66


The set of non-decomposable

-polycycles has beenclassified:

p.20/4



25/66


p.20/4



26/66


The infinite series of non-decomposable

-polycycles0

p

@

:

The only non-decomposable infinite

-polycycle are0

and0

p.20/4



27/66


The infinite series of non-decomposable generalized

-polycyclesP

Q E E !

R

% ,

,

&

:

p.20/4


28/66

III.

-maps

p.21/4

X - and d -cases


29/66

- 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


30/66

-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


31/66

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


32/66

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


33/66

case

For

&

, they exist only on sphere and are:

p.25/4

r -case


34/66

case

For

&

b

, it exist only on sphere and is:

p.25/4

r -case


35/66

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


36/66

III. Frank-Kasper maps,

i.e. -maps

p.26/4

Frank-Kasper polyhedra


37/66

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



38/66

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



39/66

y y p

A Frank-Kasper

@

-sphere p.28/4



40/66

y y p

The polycycle decomposition p.28/4



41/66

E 1

C1

C3

E

1

C3

C1

C1

E1

E1

E1

E1

Their names p.28/4



42/66

E 1

C3

E

1

E1

E1

E1

E1

C1

C1

C1

The graph of polycycles. p.28/4



43/66

E 1

C3

E

1

E1

E1

E1

E1

Q

: eliminate

m

, so as to get a

-valentmap

p.28/4

Results


44/66

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


45/66

IV.c

-maps

p.30/4

Euler formula


46/66

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



47/66

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


48/66

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


49/66

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


50/66

V. -maps

p.35/4

Euler formula


51/66

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


52/66

two possibilities (for

&

T

6

):

and the infinite series

p.37/4

W -maps y


53/66

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


54/66

III. -maps

p.39/4

Classification for W -case


55/66

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


56/66

yields

yields

yields (one infinite series)

p.41/4


57/66

yields

and

(two infinite series)

p.41/4


58/66

yields

(a family

% with@

'

)

p.41/4

W -maps r


59/66

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


60/66

III. -maps

p.43/4

Classification of W -maps e


61/66

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



62/66

Two examples amongst

T

sporadic spheres.

p.44/4



63/66

One infinite series amongst

infinite series.

p.44/4


64/66

III. -maps

p.45/4

W -case


65/66

-tori, which are

, are known for any

.For &

, they are

S .

-spheres

satisfy to ! $ &@ 3

. Is there an

infinity of such spheres?

p.46/4

W -case


66/66

The smallest

-spheres

for

&

,T

,b

are:

p.47/4

Documents

Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Polycycles and face-regular two-maps