The complete classification of lattice empty 4-simplicesÓscar Iglesias Valiño and Francisco Santos
University of Cantabria, Spain
July 31th, 2018

Workshop on Lattice Polytopes, Osaka, 2018

Oscar Iglesias

Empty 4-simplices

July 31th, 20181 / 18Empty lattice d-simplices
Definition
A lattice polytope P ⊂ Rd is a polytope with integer vertices. It is:hollow if it has no integer points in its interior.
empty (lattice-free) if it has no integer points other than its vertices.
In particular, an empty d-simplex is the convex hull of d + 1 affinely
independent integer points and not containing other integer points.

Oscar Iglesias

Empty 4-simplices

July 31th, 20182 / 18Empty lattice d-simplices
Definition
A lattice polytope P ⊂ Rd is a polytope with integer vertices. It is:hollow if it has no integer points in its interior.
empty (lattice-free) if it has no integer points other than its vertices.
In particular, an empty d-simplex is the convex hull of d + 1 affinely
independent integer points and not containing other integer points.

Empty 2 and 3-simplices and hollow 2-polytope.
Oscar Iglesias

Empty 4-simplices

July 31th, 20182 / 18Volume, width
The normalized volume Vol(P ) of a lattice polytope P equals its
Euclidean volume vol(P ) times d!.

Oscar Iglesias

Empty 4-simplices

July 31th, 20183 / 18Volume, width
The normalized volume Vol(P ) of a lattice polytope P equals its
Euclidean volume vol(P ) times d!.
It is always and integer, and for a lattice simplex∆ = conv{v1 , . . . , vd+1 }Rd it coincides with its determinant:v . . . vd+1Vol(∆) = det 1
1 ...
1

Oscar Iglesias

Empty 4-simplices

July 31th, 20183 / 18Volume, width
The normalized volume Vol(P ) of a lattice polytope P equals its
Euclidean volume vol(P ) times d!.
It is always and integer, and for a lattice simplex∆ = conv{v1 , . . . , vd+1 }Rd it coincides with its determinant:v . . . vd+1Vol(∆) = det 1
1 ...
1
The width of P ⊂ Rd with respect to a linear functional f : Rd → R
equals the difference maxx∈P f (x) − minx∈P f (x).

width(P, f ) = 4
Oscar Iglesias

Empty 4-simplices

July 31th, 20183 / 18Volume, width
The normalized volume Vol(P ) of a lattice polytope P equals its
Euclidean volume vol(P ) times d!.
It is always and integer, and for a lattice simplex∆ = conv{v1 , . . . , vd+1 }Rd it coincides with its determinant:v . . . vd+1Vol(∆) = det 1
1 ...
1
The width of P ⊂ Rd with respect to a linear functional f : Rd → R
equals the difference maxx∈P f (x) − minx∈P f (x).
We call (lattice) width of P the minimum width of P with respect tointeger functionals.

width(P ) = 2
Oscar Iglesias

Empty 4-simplices

July 31th, 20183 / 18What do we know about empty lattice d-simplices?
We write P ∼=Z Q meaning Q = φ(P ) for some unimodular affine integertransformation, φ.

Oscar Iglesias

Empty 4-simplices

July 31th, 20184 / 18What do we know about empty lattice d-simplices?
We write P ∼=Z Q meaning Q = φ(P ) for some unimodular affine integertransformation, φ. Modulo this equivalence relation:
The only empty 1-simplex is the unit segment.

Oscar Iglesias

Empty 4-simplices

July 31th, 20184 / 18What do we know about empty lattice d-simplices?
We write P ∼=Z Q meaning Q = φ(P ) for some unimodular affine integertransformation, φ. Modulo this equivalence relation:
The only empty 1-simplex is the unit segment.
The only empty 2-simplex is the unimodular triangle (' Pick’s
Theorem).

Oscar Iglesias

Empty 4-simplices

July 31th, 20184 / 18What do we know about empty lattice d-simplices?
We write P ∼=Z Q meaning Q = φ(P ) for some unimodular affine integertransformation, φ. Modulo this equivalence relation:
The only empty 1-simplex is the unit segment.
The only empty 2-simplex is the unimodular triangle (' Pick’s
Theorem).
There are infinitely many of width one (Ex: Reeve polyhedra). Empty
lattice 3-simplices are completely classified:

Theorem (White 1964)
Every empty tetrahedron of determinant q is equivalent toT (p, q) := conv{(0, 0, 0), (1, 0, 0), (0, 0, 1), (p, q, 1)}
for some p ∈ Z with gcd(p, q) = 1. Moreover, T (p, q) ∼=Z T (p0 , q) if and0±1
only if p = ±p (mod q).
Oscar Iglesias

Empty 4-simplices

July 31th, 20184 / 18What do we know about empty lattice 3-simplices
In particular, they all have width 1, i.e., they are between two parallel
lattice hyperplanes.
z(p, q, 1)

e3

z=1

y
z=0
o

e1

x

In this picture, they have width 1 with respect to the functionalf (x, y, z) = z.

Oscar Iglesias

Empty 4-simplices

July 31th, 2018

5 / 18What do we know about empty lattice 4-simplices
In contrast, a full classification of empty lattice 4-simplices is not known.
If we look at their width, we know that:

Oscar Iglesias

Empty 4-simplices

July 31th, 20186 / 18What do we know about empty lattice 4-simplices
In contrast, a full classification of empty lattice 4-simplices is not known.
If we look at their width, we know that:
1

There are infinitely many of width one.

Oscar Iglesias

Empty 4-simplices

July 31th, 20186 / 18What do we know about empty lattice 4-simplices
In contrast, a full classification of empty lattice 4-simplices is not known.
If we look at their width, we know that:
1

There are infinitely many of width one.

2

There are infinitely many of width 2 (Haase-Ziegler 2000).

Oscar Iglesias

Empty 4-simplices

July 31th, 20186 / 18What do we know about empty lattice 4-simplices
In contrast, a full classification of empty lattice 4-simplices is not known.
If we look at their width, we know that:
1

There are infinitely many of width one.

2

There are infinitely many of width 2 (Haase-Ziegler 2000).

3

The amount of empty 4-simplices of width greater than 2 is finite:

Oscar Iglesias

Empty 4-simplices

July 31th, 20186 / 18What do we know about empty lattice 4-simplices
In contrast, a full classification of empty lattice 4-simplices is not known.
If we look at their width, we know that:
1

There are infinitely many of width one.

2

There are infinitely many of width 2 (Haase-Ziegler 2000).

3

The amount of empty 4-simplices of width greater than 2 is finite:

Proposition (Blanco-Haase-Hofmann-Santos, 2016)1

2

For each d, there is a w∞ (d) such that for every n ∈ N all but finitely
many d-polytopes with n lattice points have width ≤ w∞ (d).
w∞ (4) = 2.

Oscar Iglesias

Empty 4-simplices

July 31th, 20186 / 18What do we know about empty lattice 4-simplices?
Theorem (Haase-Ziegler, 2000)
Among the 4-dimensional empty simplices with width greater than two anddeterminant D ≤ 1000,
1

All simplices of width 3 have determinant D ≤ 179, with a (unique)smallest example, of determinant D = 41, and a (unique) example ofdeterminant D = 179.

2

There is a unique class of width 4, with determinant D = 101,

3

There are no simplices of width w ≥ 5,

Oscar Iglesias

Empty 4-simplices

July 31th, 20187 / 18What do we know about empty lattice 4-simplices?
Theorem (Haase-Ziegler, 2000)
Among the 4-dimensional empty simplices with width greater than two anddeterminant D ≤ 1000,
1

All simplices of width 3 have determinant D ≤ 179, with a (unique)smallest example, of determinant D = 41, and a (unique) example ofdeterminant D = 179.

2

There is a unique class of width 4, with determinant D = 101,

3

There are no simplices of width w ≥ 5,

Conjecture (Haase-Ziegler, 2000)
The above list is complete. That is, there are no empty 4-simplices ofwidth > 2 and determinant > 179.

Theorem (I.V.-Santos, 2018)
This conjecture is true.
Oscar Iglesias

Empty 4-simplices

July 31th, 20187 / 18

Part I of the classification
Part I: Empty 4-simplices of width greater than two
The proof of the conjecture follows from the combination of atheoretical Theorem 1 and the Theorem 2 based on an enumeration:

Oscar Iglesias

Empty 4-simplices

July 31th, 20188 / 18

Part I of the classification
Part I: Empty 4-simplices of width greater than two
The proof of the conjecture follows from the combination of atheoretical Theorem 1 and the Theorem 2 based on an enumeration:

Theorem 1 (I.V.-Santos, 2018)
There is no hollow 4-simplex of width > 2 with determinant greater than
5058.

Oscar Iglesias

Empty 4-simplices

July 31th, 20188 / 18

Part I of the classification
Part I: Empty 4-simplices of width greater than two
The proof of the conjecture follows from the combination of atheoretical Theorem 1 and the Theorem 2 based on an enumeration:

Theorem 1 (I.V.-Santos, 2018)
There is no hollow 4-simplex of width > 2 with determinant greater than
5058.

Theorem 2 (I.V.-Santos, 2018)
Up to determinant ≤ 7600, all empty 4-simplices of width larger than two
have determinant in [41, 179] and are as described explicitly by Haase andZiegler.

Oscar Iglesias

Empty 4-simplices

July 31th, 20188 / 18Proof of Theorem 1
Proof of Theorem 1 relies a lot in:Convex geometry tools (succesive minima, covering minima,· · · )Oscar Iglesias

Empty 4-simplices

July 31th, 2018

9 / 18Proof of Theorem 1
Proof of Theorem 1 relies a lot in:Convex geometry tools (succesive minima, covering minima,· · · )For covering minima be here at Giulia’s talk 16:30

Oscar Iglesias

Empty 4-simplices

July 31th, 2018

9 / 18Proof of Theorem 1
Proof of Theorem 1 relies a lot in:Convex geometry tools (succesive minima, covering minima,· · · )For covering minima be here at Giulia’s talk 16:30
Classification of Maximal Hollow 3-polytopes (Averkov et al.(AWW),
2011 and Averkov et al.(AKW), 2017)

Oscar Iglesias

Empty 4-simplices

July 31th, 2018

9 / 18Proof of Theorem 1
Proof of Theorem 1 relies a lot in:Convex geometry tools (succesive minima, covering minima,· · · )For covering minima be here at Giulia’s talk 16:30
Classification of Maximal Hollow 3-polytopes (Averkov et al.(AWW),
2011 and Averkov et al.(AKW), 2017)
Get different upper bounds for the cases of P projecting to a hollow
3-polytope or not projecting to a hollow 3-polytope

Oscar Iglesias

Empty 4-simplices

July 31th, 2018

9 / 18Proof of Theorem 1
Proof of Theorem 1 relies a lot in:Convex geometry tools (succesive minima, covering minima,· · · )For covering minima be here at Giulia’s talk 16:30
Classification of Maximal Hollow 3-polytopes (Averkov et al.(AWW),
2011 and Averkov et al.(AKW), 2017)
Get different upper bounds for the cases of P projecting to a hollow
3-polytope or not projecting to a hollow 3-polytope
Indeed, no new simplex of width greater than two appear in thecomputations, so the conjecture of Haase and Ziegler is true.

Oscar Iglesias

Empty 4-simplices

July 31th, 2018

9 / 18

Hollow 3 bodies
We get a nice upper bound for the volume of hollow 3-bodies:

Theorem (I.V.-Santos, 2018)
Let w > 2.155. Then, the following statements hold for any lattice-freeconvex body K in dimension three of width at least w:
3

1

3w √
V ol(K) ≤ 4(w−(1+2/
3))

2

8w
V ol(K) ≤ (w−1)
3

3

if w ≤ 2.427,
if w ≥ 2.427.

In particular if you restrict to lattice polytopes you get part (a) of
Proposition 2 of AKW for a hollow 3 polytope Q:
vol(Q) ≤ 27

Oscar Iglesias

Empty 4-simplices

July 31th, 201810 / 18

Part II: Infinite families
Next step is classify all possible families of simplices of width one and two,and enumerate sporadic simplices that appear.

Oscar Iglesias

Empty 4-simplices

July 31th, 201811 / 18

Part II: Infinite families
Next step is classify all possible families of simplices of width one and two,and enumerate sporadic simplices that appear.
In 1988, Mori-Morrison-Morrison gave some explicit families of terminal
quotient singularities which correspond to empty 4-simplices.

Oscar Iglesias

Empty 4-simplices

July 31th, 201811 / 18

Part II: Infinite families
Next step is classify all possible families of simplices of width one and two,and enumerate sporadic simplices that appear.
In 1988, Mori-Morrison-Morrison gave some explicit families of terminal
quotient singularities which correspond to empty 4-simplices.

(Almost) Theorem 3 (Barile, Bernardi, Borisov and Kantor, 2011)
All empty 4-simplices that project to hollow 3-polytopes belong to the1 + 1 + 29 families of Mori-Morrison-Morrison (1988), all of which havewidth one or two.

Oscar Iglesias

Empty 4-simplices

July 31th, 201811 / 18

Part II: Infinite families
Next step is classify all possible families of simplices of width one and two,and enumerate sporadic simplices that appear.
In 1988, Mori-Morrison-Morrison gave some explicit families of terminal
quotient singularities which correspond to empty 4-simplices.

(Almost) Theorem 3 (Barile, Bernardi, Borisov and Kantor, 2011)
All empty 4-simplices that project to hollow 3-polytopes belong to the1 + 1 + 29 families of Mori-Morrison-Morrison (1988), all of which havewidth one or two.
But unfortunately,
Theorem 3 is only true for 4-simplices of prime volume.
With non-prime volume another 0 + 1 + 23 families arise(I.V.-Santos, 2017+).
Oscar Iglesias

Empty 4-simplices

July 31th, 201811 / 18

Classifying as projections
Theorem (Nill-Ziegler 2011)
For each d, all except finitely many hollow d-polytopes (in particular, empty
d-simplices) project to a hollow polytope of dimension < d.

Oscar Iglesias

Empty 4-simplices

July 31th, 201812 / 18

Classifying as projections
Theorem (Nill-Ziegler 2011)
For each d, all except finitely many hollow d-polytopes (in particular, empty
d-simplices) project to a hollow polytope of dimension < d.

This theorem implies that we can classify empty 4-simplices attending at
which dimension they project to:
In dimension 3 you can look at classification of hollow polytopes thisway.
Sporadic empty 4-simplices are the finitely many simplices that do not
project to a hollow polytope of dimension ≤ 3.
Empty 4-simplices of width one project to hollow 1-polytopes

Oscar Iglesias

Empty 4-simplices

July 31th, 201812 / 18

Classification of empty 4-simplices
With the new families of width two that were not described by
Mori-Morrison-Morrison, we give a complete list of families of empty
4-simplices.

Oscar Iglesias

Empty 4-simplices

July 31th, 201813 / 18

Classification of empty 4-simplices
With the new families of width two that were not described by
Mori-Morrison-Morrison, we give a complete list of families of empty
4-simplices.

Main Theorem (I.V.-Santos, ’18+)
All except finitely many empty 4-simplices belong to one of the following
cases:The three-parameter family of empty 4-simplices of width one.

Oscar Iglesias

Empty 4-simplices

July 31th, 201813 / 18

Classification of empty 4-simplices
With the new families of width two that were not described by
Mori-Morrison-Morrison, we give a complete list of families of empty
4-simplices.

Main Theorem (I.V.-Santos, ’18+)
All except finitely many empty 4-simplices belong to one of the following
cases:The three-parameter family of empty 4-simplices of width one.
Two 2-parameter families of empty 4-simplices projecting to thesecond dilation of a unimodular triangle (one listed by Mori et al., theother not).

Oscar Iglesias

Empty 4-simplices

July 31th, 201813 / 18

Classification of empty 4-simplices
With the new families of width two that were not described by
Mori-Morrison-Morrison, we give a complete list of families of empty
4-simplices.

Main Theorem (I.V.-Santos, ’18+)
All except finitely many empty 4-simplices belong to one of the following
cases:The three-parameter family of empty 4-simplices of width one.
Two 2-parameter families of empty 4-simplices projecting to thesecond dilation of a unimodular triangle (one listed by Mori et al., theother not).
The 29 Mori 1-parameter families (they project to 29 hollow"primitive" 3-polytopes).

Oscar Iglesias

Empty 4-simplices

July 31th, 201813 / 18

Classification of empty 4-simplices
With the new families of width two that were not described by
Mori-Morrison-Morrison, we give a complete list of families of empty
4-simplices.

Main Theorem (I.V.-Santos, ’18+)
All except finitely many empty 4-simplices belong to one of the following
cases:The three-parameter family of empty 4-simplices of width one.
Two 2-parameter families of empty 4-simplices projecting to thesecond dilation of a unimodular triangle (one listed by Mori et al., theother not).
The 29 Mori 1-parameter families (they project to 29 hollow"primitive" 3-polytopes).
23 additional 1-parameter families that project to 23 “non-primitive"
hollow 3-polytopes.
Oscar Iglesias

Empty 4-simplices

July 31th, 201813 / 18

Sporadic empty 4-simplices
Theorem 4 (I.V.-Santos, 2018+)
Let P be an empty 4-simplex of width two and which do not project to ahollow 3-polytope. Then Vol(P ) ≤ 7600.

Oscar Iglesias

Empty 4-simplices

July 31th, 201814 / 18

Sporadic empty 4-simplices
Theorem 4 (I.V.-Santos, 2018+)
Let P be an empty 4-simplex of width two and which do not project to ahollow 3-polytope. Then Vol(P ) ≤ 7600.
This theorem implies that no sporadic simplices will appear with volume
greater than 7600.

Theorem (I.V.-Santos, ’18+)
There are exactly 2461 (classes of) empty 4-simplices that do not belong
to any of the infinite families shown in the theorem before. These empty
4-simplices correspond to those that do not project to a hollow polytope ofdimensions d ∈ {1, 2, 3}. Their determinants range from 24 to 419.
Mori et al. present some number of sporadic simplices for prime
volume that are wrong due to some calculation mistakes
Oscar Iglesias

Empty 4-simplices

July 31th, 201814 / 18

What to look next?
Open questions and future work:
Natural question:

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

What to look next?
Open questions and future work:
Natural question: Dimension 5?

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

What to look next?
Open questions and future work:
Natural question: Dimension 5?
I

Computationally intense

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

What to look next?
Open questions and future work:
Natural question: Dimension 5?
I
I

Computationally intense
The number of families and sporadic simplices may be huge

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

What to look next?
Open questions and future work:
Natural question: Dimension 5?
Computationally intense
The number of families and sporadic simplices may be huge
I Nice properties in dimension 4, not true in dimension bigger(no cyclic group property, no unimodular facet guaranteed,...)I

I

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

What to look next?
Open questions and future work:
Natural question: Dimension 5?
Computationally intense
The number of families and sporadic simplices may be huge
I Nice properties in dimension 4, not true in dimension bigger(no cyclic group property, no unimodular facet guaranteed,...)I Maybe more interesting look first at hollow 4-polytopes instead

I

I

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

What to look next?
Open questions and future work:
Natural question: Dimension 5?
Computationally intense
The number of families and sporadic simplices may be huge
I Nice properties in dimension 4, not true in dimension bigger(no cyclic group property, no unimodular facet guaranteed,...)I Maybe more interesting look first at hollow 4-polytopes instead

I

I

What’s the maximum width for empty d-simplices depending on thedimension?

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

What to look next?
Open questions and future work:
Natural question: Dimension 5?
Computationally intense
The number of families and sporadic simplices may be huge
I Nice properties in dimension 4, not true in dimension bigger(no cyclic group property, no unimodular facet guaranteed,...)I Maybe more interesting look first at hollow 4-polytopes instead

I

I

What’s the maximum width for empty d-simplices depending on thedimension?

Oscar Iglesias

Empty 4-simplices

July 31th, 201815 / 18

Thanks for your attention!
And thank a lot Akiyoshi and Hibi
1 O. Iglesias Valiño and F. Santos, Classification of empty lattice
4-simplices of width larger than two.
To be published in TAMS
2 O. Iglesias Valiño and F. Santos, The complete classification of empty
lattice 4-simplices.
In preparation

Supported by grants MTM2017-83750-P, MTM2014-54207-P;
BES-2015-073128 of the Spanish Ministry of Science (AEI/FEDER, UE)
Oscar Iglesias

Empty 4-simplices

July 31th, 201816 / 18

Appendix
For people interested in Ehrhart polynomials:

Figure : h∗3 and h∗2 coefficients for some families of simplices
Oscar Iglesias

Empty 4-simplices

July 31th, 201817 / 18