Structure and Classifications of Lattice Polytopes Andreas Paffenholz Technische Universität Darmstadt (Classifications of) Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . . .
August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . . .
I Why? . experimentation . obtain/test conjectures
August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . . .
I Why? . experimentation . obtain/test conjectures
I How? . computational . enumerate complete subfamilies . PALP, polymake . Graded Rings Database, polymake database polyDB
August 2, 2016 | TUD | Andreas Paffenholz | 1 (Classifications of) Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I Classifications of subfamilies . by dimension . number of (interior) lattice points . structural properties: . many vertices, h∗-vector, . . .
I Why? . experimentation . obtain/test conjectures
I How? . computational . structural . enumerate complete subfamilies . standard polytope constructions . PALP, polymake . projections, liftings . Graded Rings Database, . common properties polymake database polyDB . ...
August 2, 2016 | TUD | Andreas Paffenholz | 1 Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I hyperplane description: d Q = { x | hai , xi ≤ bi } ai ∈ Z primitive, bi ∈ Z
August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I hyperplane description: d Q = { x | hai , xi ≤ bi } ai ∈ Z primitive, bi ∈ Z . assume irredundant:
Each ai defines a facet
Fi := {x ∈ Q | hai , xi = bi } of Q
August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I hyperplane description: d Q = { x | hai , xi ≤ bi } ai ∈ Z primitive, bi ∈ Z . assume irredundant:
Each ai defines a facet
Fi := {x ∈ Q | hai , xi = bi } of Q
I normal fan of Q: d ∗ . fan Σ ⊆ (R ) with rays a1, ... , am . rays form cone σ if corresponding facets define face of Q
August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes
d I lattice polytope: convex hull of finitely many points in Z d Q = conv(v1, ... , vn), vi ∈ Z
I hyperplane description: d Q = { x | hai , xi ≤ bi } ai ∈ Z primitive, bi ∈ Z . assume irredundant:
Each ai defines a facet
Fi := {x ∈ Q | hai , xi = bi } of Q
I normal fan of Q: d ∗ . fan Σ ⊆ (R ) with rays a1, ... , am . rays form cone σ if corresponding facets define face of Q
I polar (dual) polytope: Q∨ = { x | hx, vi ≤ 1 ∀v ∈ Q }
August 2, 2016 | TUD | Andreas Paffenholz | 2 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d
1 3 ehr (k) = k2 + k + 1 P 2 2
August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d
1 3 ehr (k) = k2 + k + 1 P 2 2
August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d
1 3 ehr (k) = k2 + k + 1 P 2 2
August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0
1 3 ehr (k) = k2 + k + 1 P 2 2
August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd )
1 3 ehr (k) = k2 + k + 1 P 2 2
August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd )
∗ . h0 = 1 ∗ d . h1 = |P ∩ Z | − d − 1 ∗ d . hd = | int P ∩ Z |
1 3 ehr (k) = k2 + k + 1 P 2 2
August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd )
∗ . h0 = 1 ∗ d . h1 = |P ∩ Z | − d − 1 ∗ d . hd = | int P ∩ Z |
I lattice polytope Q with normal fan Σ ←→ 1 2 3 ehrP (k) = k + k + 1 projective toric variety XΣ 2 2
with ample divisior LP
August 2, 2016 | TUD | Andreas Paffenholz | 3 Lattice Polytopes
d . Ehrhart polynomial ehrP (k) := | k · P ∩ Z | polynomial of degree d ∗ ∗ X k h (t) h -polynomial of P: ehrP (k)t = I (1 − t)d+1 k≥0 . h∗ has integral nonnegative coefficients ∗ ∗ ∗ ∗ . h -vector (h0, h1, ... , hd )
∗ . h0 = 1 ∗ d . h1 = |P ∩ Z | − d − 1 ∗ d . hd = | int P ∩ Z |
I lattice polytope Q with normal fan Σ ←→ 1 2 3 ehrP (k) = k + k + 1 projective toric variety XΣ 2 2
with ample divisior LP
. toric dictionary: properties of variety correpond to properties of polytope
August 2, 2016 | TUD | Andreas Paffenholz | 3 Classification of Lattice Polytopes
Classify or enumerate complete subfamilies of lattice polytopes
August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes
Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ...
August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes
Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ... Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
. lattice equivalence: d transformations with affine maps x 7−→ Mx + t, M unimodular, t ∈ Z .
August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes
Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ... Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
. lattice equivalence: d transformations with affine maps x 7−→ Mx + t, M unimodular, t ∈ Z .
d ∗ I P empty polytope: int P ∩ Z = ∅ ⇐⇒ hd = 0.
August 2, 2016 | TUD | Andreas Paffenholz | 4 Classification of Lattice Polytopes
Classify or enumerate complete subfamilies of lattice polytopes . by dimension . my number of (interior) lattice points . by properties of the polytopes . ... Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
. lattice equivalence: d transformations with affine maps x 7−→ Mx + t, M unimodular, t ∈ Z .
d ∗ I P empty polytope: int P ∩ Z = ∅ ⇐⇒ hd = 0. −→ different approaches for classifications of empty polytopes and those with interior lattice points.
August 2, 2016 | TUD | Andreas Paffenholz | 4 Empty Lattice Polytopes
∗ d ∗ I hd = | int P ∩ Z |, so hd = 0 iff P is empty −→ empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗)
August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes
∗ d ∗ I hd = | int P ∩ Z |, so hd = 0 iff P is empty −→ empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −→ known classifications for . polytopes for deg h∗ ∈ {0, 1} . h∗-vectors for deg h∗ = 2 . h∗-vectors with few nonzero entries
August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes
∗ d ∗ I hd = | int P ∩ Z |, so hd = 0 iff P is empty −→ empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −→ known classifications for . polytopes for deg h∗ ∈ {0, 1} . h∗-vectors for deg h∗ = 2 . h∗-vectors with few nonzero entries ∗ d . d + 1 − max(k | hk 6= 0) = min(k | int(kP) ∩ Z 6= ∅)
August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes
∗ d ∗ I hd = | int P ∩ Z |, so hd = 0 iff P is empty −→ empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −→ known classifications for . polytopes for deg h∗ ∈ {0, 1} . h∗-vectors for deg h∗ = 2 . h∗-vectors with few nonzero entries ∗ d . d + 1 − max(k | hk 6= 0) = min(k | int(kP) ∩ Z 6= ∅) I with additional conditions, small degree implies than P is a Cayley-Polytope, i.e. P splits as
P = conv( P × {ei } | i = 1, ... , k ) k for k ≥ 2 polytopes Pi and the unit vectors ei ∈ Z
. equivalently, P (lattice) projects onto a unimodular simplex of dimension at least 1
August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes
∗ d ∗ I hd = | int P ∩ Z |, so hd = 0 iff P is empty −→ empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −→ known classifications for . polytopes for deg h∗ ∈ {0, 1} . h∗-vectors for deg h∗ = 2 . h∗-vectors with few nonzero entries ∗ d . d + 1 − max(k | hk 6= 0) = min(k | int(kP) ∩ Z 6= ∅) I with additional conditions, small degree implies than P is a Cayley-Polytope, i.e. P splits as
P = conv( P × {ei } | i = 1, ... , k ) k for k ≥ 2 polytopes Pi and the unit vectors ei ∈ Z
. equivalently, P (lattice) projects onto a unimodular simplex of dimension at least 1 . general case still open
August 2, 2016 | TUD | Andreas Paffenholz | 5 Empty Lattice Polytopes
∗ d ∗ I hd = | int P ∩ Z |, so hd = 0 iff P is empty −→ empty polytopes are filtered by the degree of h∗ (number of trailing zeros in h∗) −→ known classifications for . polytopes for deg h∗ ∈ {0, 1} . h∗-vectors for deg h∗ = 2 . h∗-vectors with few nonzero entries ∗ d . d + 1 − max(k | hk 6= 0) = min(k | int(kP) ∩ Z 6= ∅) I with additional conditions, small degree implies than P is a Cayley-Polytope, i.e. P splits as
P = conv( P × {ei } | i = 1, ... , k ) k for k ≥ 2 polytopes Pi and the unit vectors ei ∈ Z
. equivalently, P (lattice) projects onto a unimodular simplex of dimension at least 1 . general case still open ∗ I intermediate zeros in h : ∗ d . h1 = |P ∩ Z | − d − 1, ∗ . h1 = 0 =⇒ P is an empty simplex
August 2, 2016 | TUD | Andreas Paffenholz | 5 Polytopes with one interior lattice point
∨ I P = Q is a Fano polytope :⇐⇒ vertices primitive, 0 ∈ int P
August 2, 2016 | TUD | Andreas Paffenholz | 6 Polytopes with one interior lattice point
∨ I P = Q is a Fano polytope :⇐⇒ vertices primitive, 0 ∈ int P
∨ I Q is reflexive :⇐⇒ Q, Q are both lattice polytopes :⇐⇒ XΣ is Gorenstein
August 2, 2016 | TUD | Andreas Paffenholz | 6 Polytopes with one interior lattice point
∨ I P = Q is a Fano polytope :⇐⇒ vertices primitive, 0 ∈ int P
∨ I Q is reflexive :⇐⇒ Q, Q are both lattice polytopes :⇐⇒ XΣ is Gorenstein −→ mirror pairs of Calabi-Yau manifolds [Batyrev, Borisov]
August 2, 2016 | TUD | Andreas Paffenholz | 6 Polytopes with one interior lattice point
∨ I P = Q is a Fano polytope :⇐⇒ vertices primitive, 0 ∈ int P
∨ I Q is reflexive :⇐⇒ Q, Q are both lattice polytopes :⇐⇒ XΣ is Gorenstein −→ mirror pairs of Calabi-Yau manifolds [Batyrev, Borisov]
I P simplicial :⇐⇒ all facets are simplices :⇐⇒ XΣ is Q-factorial
August 2, 2016 | TUD | Andreas Paffenholz | 6 Polytopes with one interior lattice point
∨ I P = Q is a Fano polytope :⇐⇒ vertices primitive, 0 ∈ int P
∨ I Q is reflexive :⇐⇒ Q, Q are both lattice polytopes :⇐⇒ XΣ is Gorenstein −→ mirror pairs of Calabi-Yau manifolds [Batyrev, Borisov]
I P simplicial :⇐⇒ all facets are simplices :⇐⇒ XΣ is Q-factorial
I P smooth :⇐⇒ vertices of facets are lattice bases :⇐⇒ XΣ is non-singular
August 2, 2016 | TUD | Andreas Paffenholz | 6 Polytopes with one interior lattice point
∨ I P = Q is a Fano polytope :⇐⇒ vertices primitive, 0 ∈ int P
∨ I Q is reflexive :⇐⇒ Q, Q are both lattice polytopes :⇐⇒ XΣ is Gorenstein −→ mirror pairs of Calabi-Yau manifolds [Batyrev, Borisov]
I P simplicial :⇐⇒ all facets are simplices :⇐⇒ XΣ is Q-factorial
I P smooth :⇐⇒ vertices of facets are lattice bases :⇐⇒ XΣ is non-singular
d I P canonical :⇐⇒ int P ∩ Z = {0} :⇐⇒ XΣ has only canonical singularities
August 2, 2016 | TUD | Andreas Paffenholz | 6 Polytopes with one interior lattice point
∨ I P = Q is a Fano polytope :⇐⇒ vertices primitive, 0 ∈ int P
∨ I Q is reflexive :⇐⇒ Q, Q are both lattice polytopes :⇐⇒ XΣ is Gorenstein −→ mirror pairs of Calabi-Yau manifolds [Batyrev, Borisov]
I P simplicial :⇐⇒ all facets are simplices :⇐⇒ XΣ is Q-factorial
I P smooth :⇐⇒ vertices of facets are lattice bases :⇐⇒ XΣ is non-singular
d I P canonical :⇐⇒ int P ∩ Z = {0} :⇐⇒ XΣ has only canonical singularities
d I P terminal :⇐⇒ P ∩ Z = {vertices} ∪ {0} :⇐⇒ XΣ has only terminal singularities
August 2, 2016 | TUD | Andreas Paffenholz | 6 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
August 2, 2016 | TUD | Andreas Paffenholz | 7 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
I computational classifications: . reflexive: 1, 16, 4319, 473800776 [Kreuzer/Skarke]
August 2, 2016 | TUD | Andreas Paffenholz | 7 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
I computational classifications: . reflexive: 1, 16, 4319, 473800776 [Kreuzer/Skarke] . terminal: 1, 5, 637 [Kasperzyk]
August 2, 2016 | TUD | Andreas Paffenholz | 7 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
I computational classifications: . reflexive: 1, 16, 4319, 473800776 [Kreuzer/Skarke] . terminal: 1, 5, 637 [Kasperzyk] . canonical: 1, 6, 674688 [Kasperzyk]
August 2, 2016 | TUD | Andreas Paffenholz | 7 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
I computational classifications: . reflexive: 1, 16, 4319, 473800776 [Kreuzer/Skarke] . terminal: 1, 5, 637 [Kasperzyk] . canonical: 1, 6, 674688 [Kasperzyk] of wich are 233 simplicial and 100 reflexive
August 2, 2016 | TUD | Andreas Paffenholz | 7 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
I computational classifications: . reflexive: 1, 16, 4319, 473800776 [Kreuzer/Skarke] . terminal: 1, 5, 637 [Kasperzyk] . canonical: 1, 6, 674688 [Kasperzyk] of wich are 233 simplicial and 100 reflexive . smooth reflexive: 1, 5, 18, 124, 866, , 7622, 72256, 749892, 8229721, | {z } |{z} | {z } | {z } [Batyrev] [Kreuzer,Nill] [Øbro] [Lorenz, P]
August 2, 2016 | TUD | Andreas Paffenholz | 7 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
I computational classifications: . reflexive: 1, 16, 4319, 473800776 [Kreuzer/Skarke] . terminal: 1, 5, 637 [Kasperzyk] . canonical: 1, 6, 674688 [Kasperzyk] of wich are 233 simplicial and 100 reflexive . smooth reflexive: 1, 5, 18, 124, 866, , 7622, 72256, 749892, 8229721, | {z } |{z} | {z } | {z } [Batyrev] [Kreuzer,Nill] [Øbro] [Lorenz, P]
. canonical/terminal polytopes can be grown from minimal ones by adding vertices
August 2, 2016 | TUD | Andreas Paffenholz | 7 Computational Classifications
Theorem (Hensley; Lagarias & Ziegler) d, m ≥ 1. Then there are, up to lattice equivalence, only finitely many d-dimensional lattice polytopes with m interior lattice points.
I Corollary number of canonical/terminal/reflexive polytopes is finite in fixed dimension
I computational classifications: . reflexive: 1, 16, 4319, 473800776 [Kreuzer/Skarke] . terminal: 1, 5, 637 [Kasperzyk] . canonical: 1, 6, 674688 [Kasperzyk] of wich are 233 simplicial and 100 reflexive . smooth reflexive: 1, 5, 18, 124, 866, , 7622, 72256, 749892, 8229721, | {z } |{z} | {z } | {z } [Batyrev] [Kreuzer,Nill] [Øbro] [Lorenz, P]
. canonical/terminal polytopes can be grown from minimal ones by adding vertices
. smooth reflexive polytopes cannot be grown from minimal ones −→ construction depends on notion of special facet and a total order on potential vertices August 2, 2016 | TUD | Andreas Paffenholz | 7 Simplicial, Terminal and Reflexive Polytopes
I structural results: terminal, canonical, or smooth lattice polytopes
August 2, 2016 | TUD | Andreas Paffenholz | 8 Simplicial, Terminal and Reflexive Polytopes
I structural results: terminal, canonical, or smooth lattice polytopes I Consider simplicial, terminal, and reflexive Polytopes (with many vertices)
August 2, 2016 | TUD | Andreas Paffenholz | 8 Simplicial, Terminal and Reflexive Polytopes
I structural results: terminal, canonical, or smooth lattice polytopes I Consider simplicial, terminal, and reflexive Polytopes (with many vertices)
I simplicial, terminal, and reflexive polytopes in low dimensions . d = 1:
August 2, 2016 | TUD | Andreas Paffenholz | 8 Simplicial, Terminal and Reflexive Polytopes
I structural results: terminal, canonical, or smooth lattice polytopes I Consider simplicial, terminal, and reflexive Polytopes (with many vertices)
I simplicial, terminal, and reflexive polytopes in low dimensions . d = 1:
. d = 2:
P6 P5 P4a P4b P3
August 2, 2016 | TUD | Andreas Paffenholz | 8 Simplicial, Terminal and Reflexive Polytopes
I structural results: terminal, canonical, or smooth lattice polytopes I Consider simplicial, terminal, and reflexive Polytopes (with many vertices)
I simplicial, terminal, and reflexive polytopes in low dimensions . d = 1:
. d = 2:
P6 P5 P4a P4b P3
e1 e1
e3 e3
. d = 3: e2 e2
−e1 −e1
August 2, 2016 | TUD | Andreas Paffenholz | 8 Simplicial, Terminal and Reflexive Polytopes
I structural results: terminal, canonical, or smooth lattice polytopes I Consider simplicial, terminal, and reflexive Polytopes (with many vertices)
I simplicial, terminal, and reflexive polytopes in low dimensions . d = 1:
. d = 2:
P6 P5 P4a P4b P3
e1 e1
e3 e3
. d = 3: e2 e2
−e1 −e1
August 2, 2016 | TUD | Andreas Paffenholz | 8 General Constructions
. P, P0 polytopes containing 0 in their interior. . direct sum P ⊕ P0 := conv ( P × {0} ∪ {0} × P0 )
. bipyramid bipyr(P) := conv({0} × P ∪ {e1, −e1}) . skew bipyramid
skewbipyr(P) := conv({0} × P ∪ {e1, v − e1}) for a vertex v of P.
August 2, 2016 | TUD | Andreas Paffenholz | 9 General Constructions
. P, P0 polytopes containing 0 in their interior. . direct sum P ⊕ P0 := conv ( P × {0} ∪ {0} × P0 )
. bipyramid bipyr(P) := conv({0} × P ∪ {e1, −e1}) . skew bipyramid
skewbipyr(P) := conv({0} × P ∪ {e1, v − e1}) for a vertex v of P.
I Proposition constructions preserve simplicial/terminal/reflexive
August 2, 2016 | TUD | Andreas Paffenholz | 9 Basic Examples
d (1) regular cross polytope: C(d) := conv( ±ei | 1 ≤ i ≤ d ) ⊂ R 0 d (2) pseudo-Del Pezzo polytope: D (d) := conv ( C(d) ∪ {1} ) ⊂ R d (3) Del Pezzo polytope: D(d) := conv ( C(d) ∪ {±1} ) ⊂ R
August 2, 2016 | TUD | Andreas Paffenholz | 10 Basic Examples
d (1) regular cross polytope: C(d) := conv( ±ei | 1 ≤ i ≤ d ) ⊂ R 0 d (2) pseudo-Del Pezzo polytope: D (d) := conv ( C(d) ∪ {1} ) ⊂ R d (3) Del Pezzo polytope: D(d) := conv ( C(d) ∪ {±1} ) ⊂ R
I Theorem [Voskresenskii&Klyachko, Ewald, Nill] P simplicial, terminal, and reflexive with antipodal pair of facets =⇒ P is direct sum of a centrally symmetric cross polytope, (2), and (3)
August 2, 2016 | TUD | Andreas Paffenholz | 10 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
P6
P5
August 2, 2016 | TUD | Andreas Paffenholz | 11 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
Theorem There are no other cases. [Casagrande] I P6
P5
August 2, 2016 | TUD | Andreas Paffenholz | 11 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
Theorem There are no other cases. [Casagrande] I P6
⊕d/2−1 I f0 = 3d − 1: (b) P5 ⊕ P6 ⊕(d−1)/2 (c) proper or skew bipyramid over P6
P5
August 2, 2016 | TUD | Andreas Paffenholz | 11 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
Theorem There are no other cases. [Casagrande] I P6
⊕d/2−1 I f0 = 3d − 1: (b) P5 ⊕ P6 ⊕(d−1)/2 (c) proper or skew bipyramid over P6
I Theorem There are no other cases. [Øbro & Nill] P5
August 2, 2016 | TUD | Andreas Paffenholz | 11 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
Theorem There are no other cases. [Casagrande] I P6
⊕d/2−1 I f0 = 3d − 1: (b) P5 ⊕ P6 ⊕(d−1)/2 (c) proper or skew bipyramid over P6
I Theorem There are no other cases. [Øbro & Nill] P5
2 ⊕d/2−2 I f0 = 3d − 2: (d) P5 ⊕ P6 ⊕d/2−2 (e) D(4) ⊕ P6 (f) double proper or skew bipyramid over (b) or (c) (g) double proper or skew bipyramid over (a)
August 2, 2016 | TUD | Andreas Paffenholz | 11 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
Theorem There are no other cases. [Casagrande] I P6
⊕d/2−1 I f0 = 3d − 1: (b) P5 ⊕ P6 ⊕(d−1)/2 (c) proper or skew bipyramid over P6
I Theorem There are no other cases. [Øbro & Nill] P5
2 ⊕d/2−2 I f0 = 3d − 2: (d) P5 ⊕ P6 ⊕d/2−2 (e) D(4) ⊕ P6 (f) double proper or skew bipyramid over (b) or (c) (g) double proper or skew bipyramid over (a)
I Theorem There are no other cases. [Assarf, Joswig, P]
August 2, 2016 | TUD | Andreas Paffenholz | 11 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
Theorem There are no other cases. [Casagrande] I P6
⊕d/2−1 I f0 = 3d − 1: (b) P5 ⊕ P6 ⊕(d−1)/2 (c) proper or skew bipyramid over P6
I Theorem There are no other cases. [Øbro & Nill] P5
2 ⊕d/2−2 I f0 = 3d − 2: (d) P5 ⊕ P6 ⊕d/2−2 (e) D(4) ⊕ P6 (f) double proper or skew bipyramid over (b) or (c) (g) double proper or skew bipyramid over (a)
I Theorem There are no other cases. [Assarf, Joswig, P]
August 2, 2016 | TUD | Andreas Paffenholz | 11 Many Vertices
⊕d/2 I f0 = 3d: (a) P6
Theorem There are no other cases. [Casagrande] I P6
⊕d/2−1 I f0 = 3d − 1: (b) P5 ⊕ P6 ⊕(d−1)/2 (c) proper or skew bipyramid over P6
I Theorem There are no other cases. [Øbro & Nill] P5
2 ⊕d/2−2 I f0 = 3d − 2: (d) P5 ⊕ P6 ⊕d/2−2 (e) D(4) ⊕ P6 (f) double proper or skew bipyramid over (b) or (c) (g) double proper or skew bipyramid over (a)
I Theorem There are no other cases. [Assarf, Joswig, P]
August 2, 2016 | TUD | Andreas Paffenholz | 11 η-vectors
I P simplicial, terminal, and reflexive d-polytope,
. F a facet of P with normal uF
−→ given by primitive facet normal uF
F = {x ∈ P | h uF , x i = 1 }
August 2, 2016 | TUD | Andreas Paffenholz | 12 η-vectors
I P simplicial, terminal, and reflexive d-polytope,
. F a facet of P with normal uF
−→ given by primitive facet normal uF
F = {x ∈ P | h uF , x i = 1 }
−→ uF induces grading on V(P) by distance from F
August 2, 2016 | TUD | Andreas Paffenholz | 12 η-vectors
I P simplicial, terminal, and reflexive d-polytope,
. F a facet of P with normal uF
−→ given by primitive facet normal uF
F = {x ∈ P | h uF , x i = 1 }
−→ uF induces grading on V(P) by distance from F F −→ η-vector η = (η1, η0, η−1, ... ) ,
ηi := |{ x ∈ V(P) | h u, x i = i }|
August 2, 2016 | TUD | Andreas Paffenholz | 12 η-vectors
I P simplicial, terminal, and reflexive d-polytope,
. F a facet of P with normal uF
−→ given by primitive facet normal uF
F = {x ∈ P | h uF , x i = 1 }
−→ uF induces grading on V(P) by distance from F F −→ η-vector η = (η1, η0, η−1, ... ) ,
ηi := |{ x ∈ V(P) | h u, x i = i }| −→ Partition vertex set V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
August 2, 2016 | TUD | Andreas Paffenholz | 12 η-vectors
I P simplicial, terminal, and reflexive d-polytope,
. F a facet of P with normal uF
−→ given by primitive facet normal uF
F = {x ∈ P | h uF , x i = 1 }
−→ uF induces grading on V(P) by distance from F F −→ η-vector η = (η1, η0, η−1, ... ) ,
ηi := |{ x ∈ V(P) | h u, x i = i }| −→ Partition vertex set V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I F is a special facet P :⇐⇒ vP := v∈V(P) v ∈ cone(F)
August 2, 2016 | TUD | Andreas Paffenholz | 12 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
. equality: V (F) − {ui } + {x} is facet
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
. equality: V (F) − {ui } + {x} is facet
. Vertices in V (F, 0) are on facets adjacent to F
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
. equality: V (F) − {ui } + {x} is facet
. Vertices in V (F, 0) are on facets adjacent to F
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
. equality: V (F) − {ui } + {x} is facet
. Vertices in V (F, 0) are on facets adjacent to F
I Proposition η0 ≤ d [Nill]
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
. equality: V (F) − {ui } + {x} is facet
. Vertices in V (F, 0) are on facets adjacent to F
I Proposition η0 ≤ d [Nill]
I Proposition η0 ≥ d − 1 =⇒ u1, ... , ud are lattice basis
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
. equality: V (F) − {ui } + {x} is facet
. Vertices in V (F, 0) are on facets adjacent to F
I Proposition η0 ≤ d [Nill]
I Proposition η0 ≥ d − 1 =⇒ u1, ... , ud are lattice basis
I Theorem f0 := | V(P)| ≤ 3d [Casagrande; Øbro]
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
. Fix special facet F,
. vertices {ui }, dual basis {ˆui } . V (P) := V (F) ∪ V (F, 0) ∪ V (F, −1) ∪ ...
I Proposition [Øbro] . Coordinates of vertices are bounded in dual basis
. x ∈ V (F, k) =⇒ hˆui , xi ≥ k − 1
. equality: V (F) − {ui } + {x} is facet
. Vertices in V (F, 0) are on facets adjacent to F
I Proposition η0 ≤ d [Nill]
I Proposition η0 ≥ d − 1 =⇒ u1, ... , ud are lattice basis
I Theorem f0 := | V(P)| ≤ 3d [Casagrande; Øbro] P proof: 0 ≤ h uF , v∈V (P) v i = η1 + 0 · η0 + (−1) · η−1 + (−2) · η−2 + ··· = d + 0 − · · ·
August 2, 2016 | TUD | Andreas Paffenholz | 13 Many Vertices
I classify η-vectors for a special facet F P I vP := v∈V(P) v
I `: height of vP above F
August 2, 2016 | TUD | Andreas Paffenholz | 14 Many Vertices
I classify η-vectors for a special facet F ` 2 1 1 0 0 0 0
P η1 d d d d d d d I vP := v∈V(P) v η0 d d d − 1 d d d − 1 d − 2 η 2 3 1 3 4 2 I `: height of vP above F −1 d − d − d − d − d − d − d η−2 0 1 0 0 2 1 0 η−3 0 0 0 1 0 0 0
(a) (b) (c) (d) (e) (f) (g) | {z } all facets special
August 2, 2016 | TUD | Andreas Paffenholz | 14 Many Vertices
I classify η-vectors for a special facet F ` 2 1 1 0 0 0 0
P η1 d d d d d d d I vP := v∈V(P) v η0 d d d − 1 d d d − 1 d − 2 η 2 3 1 3 4 2 I `: height of vP above F −1 d − d − d − d − d − d − d η−2 0 1 0 0 2 1 0 η−3 0 0 0 1 0 0 0
(a) (b) (c) (d) (e) (f) (g) | {z } all facets special
I consider cases separately
August 2, 2016 | TUD | Andreas Paffenholz | 14 Many Vertices
I classify η-vectors for a special facet F ` 2 1 1 0 0 0 0
P η1 d d d d d d d I vP := v∈V(P) v η0 d d d − 1 d d d − 1 d − 2 η 2 3 1 3 4 2 I `: height of vP above F −1 d − d − d − d − d − d − d η−2 0 1 0 0 2 1 0 η−3 0 0 0 1 0 0 0
(a) (b) (c) (d) (e) (f) (g) | {z } all facets special
I consider cases separately . e.g., . all ηF of type (g) =⇒ polytope is centrally symmetric
August 2, 2016 | TUD | Andreas Paffenholz | 14 Many Vertices
I classify η-vectors for a special facet F ` 2 1 1 0 0 0 0
P η1 d d d d d d d I vP := v∈V(P) v η0 d d d − 1 d d d − 1 d − 2 η 2 3 1 3 4 2 I `: height of vP above F −1 d − d − d − d − d − d − d η−2 0 1 0 0 2 1 0 η−3 0 0 0 1 0 0 0
(a) (b) (c) (d) (e) (f) (g) | {z } all facets special
I consider cases separately . e.g., . all ηF of type (g) =⇒ polytope is centrally symmetric . (d) does not occur ←− look at adjacent facet
August 2, 2016 | TUD | Andreas Paffenholz | 14 Many Vertices
I classify η-vectors for a special facet F ` 2 1 1 0 0 0 0
P η1 d d d d d d d I vP := v∈V(P) v η0 d d d − 1 d d d − 1 d − 2 η 2 3 1 3 4 2 I `: height of vP above F −1 d − d − d − d − d − d − d η−2 0 1 0 0 2 1 0 η−3 0 0 0 1 0 0 0
(a) (b) (c) (d) (e) (f) (g) | {z } all facets special
I consider cases separately . e.g., . all ηF of type (g) =⇒ polytope is centrally symmetric . (d) does not occur ←− look at adjacent facet
I show that polytopes are either . direct sum of P6 with (d − 2)-polytope . (skew) bipyramid over (d − 1)-polytope
August 2, 2016 | TUD | Andreas Paffenholz | 14 Many Vertices
I classify η-vectors for a special facet F ` 2 1 1 0 0 0 0
P η1 d d d d d d d I vP := v∈V(P) v η0 d d d − 1 d d d − 1 d − 2 η 2 3 1 3 4 2 I `: height of vP above F −1 d − d − d − d − d − d − d η−2 0 1 0 0 2 1 0 η−3 0 0 0 1 0 0 0
(a) (b) (c) (d) (e) (f) (g) | {z } all facets special
I consider cases separately . e.g., . all ηF of type (g) =⇒ polytope is centrally symmetric . (d) does not occur ←− look at adjacent facet
I show that polytopes are either . direct sum of P6 with (d − 2)-polytope . (skew) bipyramid over (d − 1)-polytope I exact types via induction
August 2, 2016 | TUD | Andreas Paffenholz | 14 Can we continue?
e1+e3
I f0 = 3d − 3 ? . R := skew bipyramid over P6 e3 −→ 8 vertices and 12 facets e . P := R⊕3 2 −→ 3 · 8 = 3 · 9 − 3 vertices in dimension d = 9
. P is not a (skew) bipyramid over a sum of P5 and P6 −e1
August 2, 2016 | TUD | Andreas Paffenholz | 15 Can we continue?
e1+e3
I f0 = 3d − 3 ? . R := skew bipyramid over P6 e3 −→ 8 vertices and 12 facets e . P := R⊕3 2 −→ 3 · 8 = 3 · 9 − 3 vertices in dimension d = 9
. P is not a (skew) bipyramid over a sum of P5 and P6 −e1 I Theorem [Assarf, Joswig, P] P terminal, simplicial, reflexive d-polytope with 3d − 2 vertices 2 ⊕d/2−2 Then P is . P5 ⊕ P6 , or ⊕d/2−2 . D(4) ⊕ P6 , or ⊕k . (double) proper/skew bipyramid over P6 for suitable k
August 2, 2016 | TUD | Andreas Paffenholz | 15 Can we continue?
e1+e3
I f0 = 3d − 3 ? . R := skew bipyramid over P6 e3 −→ 8 vertices and 12 facets e . P := R⊕3 2 −→ 3 · 8 = 3 · 9 − 3 vertices in dimension d = 9
. P is not a (skew) bipyramid over a sum of P5 and P6 −e1 I Theorem [Assarf, Joswig, P] P terminal, simplicial, reflexive d-polytope with 3d − 2 vertices ⊕k Then P = Q ⊕ P6 for suitable k and dim Q ≤ 4.
August 2, 2016 | TUD | Andreas Paffenholz | 15 Can we continue?
e1+e3
I f0 = 3d − 3 ? . R := skew bipyramid over P6 e3 −→ 8 vertices and 12 facets e . P := R⊕3 2 −→ 3 · 8 = 3 · 9 − 3 vertices in dimension d = 9
. P is not a (skew) bipyramid over a sum of P5 and P6 −e1 I Theorem [Assarf, Joswig, P] P terminal, simplicial, reflexive d-polytope with 3d − 2 vertices ⊕k Then P = Q ⊕ P6 for suitable k and dim Q ≤ 4.
I Conjecture [Assarf, Joswig, P] P smooth Fano d-polytope with 3d − k vertices, k ≤ d/3 l Then P = Q ⊕ P6 for dim Q ≤ 3k and appropriate l.
August 2, 2016 | TUD | Andreas Paffenholz | 15 Can we continue?
e1+e3
I f0 = 3d − 3 ? . R := skew bipyramid over P6 e3 −→ 8 vertices and 12 facets e . P := R⊕3 2 −→ 3 · 8 = 3 · 9 − 3 vertices in dimension d = 9
. P is not a (skew) bipyramid over a sum of P5 and P6 −e1 I Theorem [Assarf, Joswig, P] P terminal, simplicial, reflexive d-polytope with 3d − 2 vertices ⊕k Then P = Q ⊕ P6 for suitable k and dim Q ≤ 4.
I Conjecture [Assarf, Joswig, P] P smooth Fano d-polytope with 3d − k vertices, k ≤ d/3 l Then P = Q ⊕ P6 for dim Q ≤ 3k and appropriate l.
. weak version of conjecture is true: I Theorem [Assarf, Nill] For sufficiently large d, v a smooth Fano d-polytope with v vertices has a P6-factor.
August 2, 2016 | TUD | Andreas Paffenholz | 15 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
. Consider convex hull S of [ 0, 0, 0 ], [ 1, 1, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ]
−→ S is lattice simplex of volume and facet width 2
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
. Consider convex hull S of [ 0, 0, 0 ], [ 1, 1, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ]
−→ S is lattice simplex of volume and facet width 2
I Let P := conv ( S × {1}, −S × {−1} )
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
. Consider convex hull S of [ 0, 0, 0 ], [ 1, 1, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ]
−→ S is lattice simplex of volume and facet width 2
I Let P := conv ( S × {1}, −S × {−1} ) . P is simplicial, terminal, reflexive
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
. Consider convex hull S of [ 0, 0, 0 ], [ 1, 1, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ]
−→ S is lattice simplex of volume and facet width 2
I Let P := conv ( S × {1}, −S × {−1} ) . P is simplicial, terminal, reflexive . vertices of facet S are not a lattice basis
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
. Consider convex hull S of [ 0, 0, 0 ], [ 1, 1, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ]
−→ S is lattice simplex of volume and facet width 2
I Let P := conv ( S × {1}, −S × {−1} ) . P is simplicial, terminal, reflexive . vertices of facet S are not a lattice basis . P has 12 = 3d − 4 vertices
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
. Consider convex hull S of [ 0, 0, 0 ], [ 1, 1, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ]
−→ S is lattice simplex of volume and facet width 2
I Let P := conv ( S × {1}, −S × {−1} ) . P is simplicial, terminal, reflexive . vertices of facet S are not a lattice basis . P has 12 = 3d − 4 vertices
−→ For f0 ≤ 3d − 4 there are nonsmooth simplicial, terminal, and reflexive polytopes
August 2, 2016 | TUD | Andreas Paffenholz | 16 STR does not imply smoothness
I A posteriori: All simplicial, terminal, reflexive polytopes with at least f0 := 3d − 2 vertices are (dual to) smooth
I Not true in general [Haase]
. Consider convex hull S of [ 0, 0, 0 ], [ 1, 1, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ]
−→ S is lattice simplex of volume and facet width 2
I Let P := conv ( S × {1}, −S × {−1} ) . P is simplicial, terminal, reflexive . vertices of facet S are not a lattice basis . P has 12 = 3d − 4 vertices
−→ For f0 ≤ 3d − 4 there are nonsmooth simplicial, terminal, and reflexive polytopes
I open case: simplicial, terminal, reflexive polytopes with 3d − 3 vertices
August 2, 2016 | TUD | Andreas Paffenholz | 16 polymake
I polymake: software framework for computations in discrete geometry, toric geometry, tropical geometry . interactive . rule based . easy extensions current version: 3.02, Linux/Mac OS, written in perl, C++, Java founded by Michael Joswig (TU Berlin), Ewgenij Gawrilow (TomTom) available at polymake.org, GPL licensed
August 2, 2016 | TUD | Andreas Paffenholz | 17 polymake
I polymake: software framework for computations in discrete geometry, toric geometry, tropical geometry . interactive . rule based . easy extensions current version: 3.02, Linux/Mac OS, written in perl, C++, Java founded by Michael Joswig (TU Berlin), Ewgenij Gawrilow (TomTom) available at polymake.org, GPL licensed
I polyDB: database extension for polymake . direct access from polymake . independent access/access from other software possible . web based interface (planned) beta version, developed by: Silke Horn (iteratec), P. available at github.org/solros/poly_db, GPL licensed
August 2, 2016 | TUD | Andreas Paffenholz | 17