many geometric realizations of graph associahedra€¦ · geometric realizations of graph...
TRANSCRIPT
Sage Days 64.5
June 3, 2015
MANYGEOMETRIC
REALIZATIONSOF GRAPH
ASSOCIAHEDRAT. MANNEVILLE V. PILAUD
(LIX) (CNRS & LIX)
POLYTOPES & COMBINATORICS
SIMPLICIAL COMPLEX
simplicial complex = collection of subsets of X downward closed
exm:
X = [n] ∪ [n]
∆ = {I ⊆ X | ∀i ∈ [n], {i, i} 6⊆ I}12 13 13 12 23 23 23 23 12 13 13 12
1 2 3 3 2 1
123 123 123 123 123 123 123 123
FANS
polyhedral cone = positive span of a finite set of Rd
= intersection of finitely many linear half-spaces
fan = collection of polyhedral cones closed by faces
and where any two cones intersect along a face
simplicial fan = maximal cones generated by d rays
POLYTOPES
polytope = convex hull of a finite set of Rd
= bounded intersection of finitely many affine half-spaces
face = intersection with a supporting hyperplane
face lattice = all the faces with their inclusion relations
simple polytope = facets in general position = each vertex incident to d facets
SIMPLICIAL COMPLEXES, FANS, AND POLYTOPES
P polytope, F face of P
normal cone of F = positive span of the outer normal vectors of the facets containing F
normal fan of P = { normal cone of F | F face of P }
simple polytope =⇒ simplicial fan =⇒ simplicial complex
PERMUTAHEDRON
34123421
43214312
2413
4213
3214
14231432
1342
12431234
21341324
2341
2431
31242314
Permutohedron Perm(n)
= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}
= H ∩⋂
∅6=J([n+1]
{x ∈ Rn+1
∣∣∣∣ ∑j∈J
xj ≥(|J | + 1
2
)}
PERMUTAHEDRON
34123421
43214312
2413
4213
3214
14231432
1342
12431234
21341324
2341
2431
31242314
Permutohedron Perm(n)
= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}
= H ∩⋂
∅6=J([n+1]
{x ∈ Rn+1
∣∣∣∣ ∑j∈J
xj ≥(|J | + 1
2
)}connections to
• weak order
• reduced expressions
• braid moves
• cosets of the symmetric group
PERMUTAHEDRON
34123421
43214312
2413
4213
3214
14231432
1342
12431234
21341324
2341
2431
31242314
1211
1221
1222
1212
2212
1112
2211
2321
13212331
12311332
1322
1232
1233
1323
1223
1312 2313
1213
2113
32132312
32122311
32113321
1123
2123
3312
Permutohedron Perm(n)
= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}
= H ∩⋂
∅6=J([n+1]
{x ∈ Rn+1
∣∣∣∣ ∑j∈J
xj ≥(|J | + 1
2
)}connections to
• weak order
• reduced expressions
• braid moves
• cosets of the symmetric group
k-faces of Perm(n)≡ surjections from [n + 1]
to [n + 1− k]
PERMUTAHEDRON
3|4|1|24|3|1|2
4|3|2|13|4|2|1
3|1|4|2
3|2|4|1
3|2|1|4
1|3|4|21|4|3|2
1|4|2|3
1|2|4|31|2|3|4
2|1|3|41|3|2|4
4|1|2|3
4|1|3|2
2|3|1|43|1|2|4
134|2
14|23
1|234
13|24
3|124
123|4
34|12
4|13|2
14|3|24|1|23
14|2|31|4|23
1|24|3
13|2|4
1|23|4
13|4|2 3|1|24
13|2|4
23|1|4
3|2|143|14|2
3|24|1
12|3|41|2|34
2|13|4
34|2|13|4|1234|1|2
4|3|12
Permutohedron Perm(n)
= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}
= H ∩⋂
∅6=J([n+1]
{x ∈ Rn+1
∣∣∣∣ ∑j∈J
xj ≥(|J | + 1
2
)}connections to
• weak order
• reduced expressions
• braid moves
• cosets of the symmetric group
k-faces of Perm(n)≡ surjections from [n + 1]
to [n + 1− k]
≡ ordered partitions of [n + 1]
into n + 1− k parts
PERMUTAHEDRON
3|4|1|24|3|1|2
4|3|2|13|4|2|1
3|1|4|2
3|2|4|1
3|2|1|4
1|3|4|21|4|3|2
1|4|2|3
1|2|4|31|2|3|4
2|1|3|41|3|2|4
4|1|2|3
4|1|3|2
2|3|1|43|1|2|4
134|2
14|23
1|234
13|24
3|124
123|4
34|12
4|13|2
14|3|24|1|23
14|2|31|4|23
1|24|3
13|2|4
1|23|4
13|4|2 3|1|24
13|2|4
23|1|4
3|2|143|14|2
3|24|1
12|3|41|2|34
2|13|4
34|2|13|4|1234|1|2
4|3|12
Permutohedron Perm(n)
= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}
= H ∩⋂
∅6=J([n+1]
{x ∈ Rn+1
∣∣∣∣ ∑j∈J
xj ≥(|J | + 1
2
)}connections to
• weak order
• reduced expressions
• braid moves
• cosets of the symmetric group
k-faces of Perm(n)≡ surjections from [n + 1]
to [n + 1− k]
≡ ordered partitions of [n + 1]
into n + 1− k parts
≡ collections of n− k nested
subsets of [n + 1]
COXETER ARRANGEMENT
34|12
134|23|124
13|24
123|41|234
14|23
3|4|1|24|3|1|2
4|3|2|1 3|4|2|1
3|1|4|2
3|2|4|1
3|2|1|41|3|4|21|4|3|2
1|4|2|3
1|2|4|31|2|3|4
2|1|3|4
1|3|2|4
4|1|2|3
4|1|3|2
2|3|1|4
3|1|2|4
Coxeter fan
= fan defined by the hyperplane arrangement{x ∈ Rn+1
∣∣ xi = xj}1≤i<j≤n+1
= collection of all cones{x ∈ Rn+1
∣∣ xi < xj if π(i) < π(j)}
for all surjections π : [n + 1]→ [n + 1− k]
(n− k)-dimensional cones
≡ surjections from [n + 1]
to [n + 1− k]
≡ ordered partitions of [n + 1]
into n + 1− k parts
≡ collections of n− k nested
subsets of [n + 1]
ASSOCIAHEDRA
ASSOCIAHEDRON
Associahedron = polytope whose face lattice is isomorphic to the lattice of crossing-free
sets of internal diagonals of a convex (n + 3)-gon, ordered by reverse inclusion
vertices ↔ triangulations
edges ↔ flips
faces ↔ dissections
vertices ↔ binary trees
edges ↔ rotations
faces ↔ Schroder trees
VARIOUS ASSOCIAHEDRA
Associahedron = polytope whose face lattice is isomorphic to the lattice of crossing-free
sets of internal diagonals of a convex (n + 3)-gon, ordered by reverse inclusion
(Pictures by Ceballos-Santos-Ziegler)Tamari (’51) — Stasheff (’63) — Haimann (’84) — Lee (’89) —
. . . — Gel’fand-Kapranov-Zelevinski (’94) — . . . — Chapoton-Fomin-Zelevinsky (’02) — . . . — Loday (’04) — . . .
— Ceballos-Santos-Ziegler (’11)
THREE FAMILIES OF REALIZATIONS
SECONDARY
POLYTOPE
Gelfand-Kapranov-Zelevinsky (’94)
Billera-Filliman-Sturmfels (’90)
LODAY’S
ASSOCIAHEDRON
Loday (’04)
Hohlweg-Lange (’07)
Hohlweg-Lange-Thomas (’12)
CHAP.-FOM.-ZEL.’S
ASSOCIAHEDRON
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@@@@BBBBBB
@@
@@@
@@@
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p p p p p p p pp p p p p p p p
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pppppppppppppppppppppppppppppppppppppppppα2
α1+α2
α2+α3
α1+α2+α3
α1
α3
(Pictures by CFZ)
Chapoton-Fomin-Zelevinsky (’02)
Ceballos-Santos-Ziegler (’11)
THREE FAMILIES OF REALIZATIONS
SECONDARY
POLYTOPE
Gelfand-Kapranov-Zelevinsky (’94)
Billera-Filliman-Sturmfels (’90)
LODAY’S
ASSOCIAHEDRON
Loday (’04)
Hohlweg-Lange (’07)
Hohlweg-Lange-Thomas (’12)
Hopf
algebra
Cluster
algebras
CHAP.-FOM.-ZEL.’S
ASSOCIAHEDRON
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@@@@
�������
�����
����
@@@@BBBBBB
@@
@@@
@@@
��
��
p p p p p p p pp p p p p p p p
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pppppppppppppppppppppppppppppppppppppppppα2
α1+α2
α2+α3
α1+α2+α3
α1
α3
(Pictures by CFZ)
Chapoton-Fomin-Zelevinsky (’02)
Ceballos-Santos-Ziegler (’11)
Cluster
algebras
��������
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@@@
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������
@@@
HHH
HH
���BBBB
HHH
@@
@@
@@@
HHH
���A
AA�� ��
HHHHH
p p p p p p pp p p p p p p
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p ppppppppppppppppppppppppppα2
α1+α2α2+α3
2α2 + α3
α1
α3
2α1 + 2α2 + α3
GRAPH ASSOCIAHEDRA
NESTED COMPLEX AND GRAPH ASSOCIAHEDRON
G graph on ground set V
Tube of G = connected induced subgraph of G
Compatible tubes = nested, or disjoint and non-adjacent
Tubing on G = collection of pairwise compatible tubes of G
24 5
3
1 2 3
6
87 9
24 5
3
1 2 3
6
87 9
Nested complex N (G) = simplicial complex of tubings on G
= clique complex of the compatibility relation on tubes
G-associahedron = polytopal realization of the nested complex on G
Carr-Devadoss, Coxeter complexes and graph associahedra (’06)
EXM: NESTED COMPLEX
EXM: GRAPH ASSOCIAHEDRON
SPECIAL GRAPH ASSOCIAHEDRA
Path associahedron Cycle associahedron Complete graph associahedron
= associahedron = cyclohedron = permutahedron
341234214321 4312
2413
4213
3214
14231432
1342
1243 12342134
1324
2341
2431
31242314
LINEAR LAURENT PHENOMENON ALGEBRAS
Laurent Phenomenon Algebra = commut. ring gen. by cluster variables grouped in clusters
seed = pair (x,F) where
• x = {x1, . . . , xn} cluster variables
• F = {F1, . . . , Fn} exchange polynomials
seed mutation = (x,F) 7−→ µi(x,F) = (x′,F′) where
• x′i = Fi/xi while x′j = xj for j 6= i
• F ′j obtained from Fj by eliminating xi
THM. Every cluster variable in a LP algebra is a Laurent polynomial in the cluster
variables of any seed.Lam-Pylyavskyy, Laurent Phenomenon Algebras (’12)
Connection to graph associahedra: Any (directed) graph G defines a linear LP algebra
whose cluster complex contains the nested complex of G
Lam-Pylyavskyy, Linear Laurent Phenomenon Algebras (’12)
COMPATIBILITY FANSFOR GRAPHICAL NESTED COMPLEXES
Thibault Manneville & VP
arXiv:1501.07152
COMPATIBILITY FANS FOR ASSOCIAHEDRA
T◦ an initial triangulation
δ, δ′ two internal diagonals
compatibility degree between δ and δ′
(δ ‖ δ′) =
−1 if δ = δ′
0 if δ and δ′ do not cross1 if δ and δ′ cross
compatibility vector of δ wrt T◦:
d(T◦, δ) =[(δ◦ ‖ δ)
]δ◦∈T◦
compatibility fan wrt T◦
D(T◦) = {R≥0 d(T◦,D) | D dissection}
Fomin-Zelevinsky, Y -Systems and generalized associahedra (’03)
Fomin-Zelevinsky, Cluster algebras II: Finite type classification (’03)
Chapoton-Fomin-Zelevinsky, Polytopal realizations of generalized associahedra (’02)
Ceballos-Santos-Ziegler, Many non-equivalent realizations of the associahedron (’11)
COMPATIBILITY FANS FOR ASSOCIAHEDRA
Different initial triangulations T◦ yield different realizations
THM. For any initial triangulation T◦, the cones {R≥0 d(T◦,D) | D dissection} form a
complete simplicial fan. Moreover, this fan is always polytopal.
Ceballos-Santos-Ziegler, Many non-equivalent realizations of the associahedron (’11)
COMPATIBILITY FANS FOR GRAPHICAL NESTED COMPLEXES
T◦ an initial maximal tubing on G
t, t′ two tubes of G
compatibility degree between t and t′
(t ‖ t′) =
−1 if t = t′
0 if t, t′ are compatible|{neighbors of t in t′r t}| otherwise
compatibility vector of t wrt T◦:
d(T◦, t) =[(t◦ ‖ t)
]t◦∈T◦
THM. For any initial maximal tubing T◦ on G,
the collection of cones
D(G,T◦) = {R≥0 d(T◦,T) | T tubing on G}
forms a complete simplicial fan, called com-
patibility fan of G.
Manneville-P., Compatibility fans for graphical nested complexes
GRAPH CATALAN MANY SIMPLICIAL FAN REALIZATIONS
THM. When none of the connected components of G is a spider,
# linear isomorphism classes of compatibility fans of G
= # orbits of maximal tubings on G under graph automorphisms of G.
Manneville-P., Compatibility fans for graphical nested complexes (’15)
POLYTOPALITY?
QU. Are all compatibility fans polytopal?
POLYTOPALITY?
QU. Are all compatibility fans polytopal?
Polytopality of a complete simplicial fan ⇐⇒ Feasibility of a linear program
Exm: We check that the compatibility fan on the complete graph K7 is polytopal by
solving a linear program on 126 variables and 17 640 inequalities
POLYTOPALITY?
QU. Are all compatibility fans polytopal?
Polytopality of a complete simplicial fan ⇐⇒ Feasibility of a linear program
Exm: We check that the compatibility fan on the complete graph K7 is polytopal by
solving a linear program on 126 variables and 17 640 inequalities
=⇒ All compatibility fans on complete graphs of ≤ 7 vertices are polytopal...
=⇒ All compatibility fans on graphs of ≤ 4 vertices are polytopal...
POLYTOPALITY?
QU. Are all compatibility fans polytopal?
Polytopality of a complete simplicial fan ⇐⇒ Feasibility of a linear program
Exm: We check that the compatibility fan on the complete graph K7 is polytopal by
solving a linear program on 126 variables and 17 640 inequalities
=⇒ All compatibility fans on complete graphs of ≤ 7 vertices are polytopal...
=⇒ All compatibility fans on graphs of ≤ 4 vertices are polytopal...
To go further, we need to understand better the linear dependences between the compat-
ibility vectors of the tubes involved in a flip
THM. All compatibility fans on the paths and cycles are polytopal
Ceballos-Santos-Ziegler, Many non-equivalent realizations of the associahedron (’11)
Manneville-P., Compatibility fans for graphical nested complexes (’15)
POLYTOPALITY?
QU. Are all compatibility fans polytopal?
Remarkable realizations of the stellohedra
POLYTOPALITY?
QU. Are all compatibility fans polytopal?
Remarkable realizations of the stellohedra
Convex hull of the orbits under coordinate permutations of the set{∑
i>k i ei∣∣ 0 ≤ k ≤ n
}
LODAY’S ASSOCIAHEDRON
Asso(n) := conv {L(T) | T binary tree} = H ∩⋂
1≤i≤j≤n+1
H≥(i, j)
L(T) :=[`(T, i) · r(T, i)
]i∈[n+1]
H≥(i, j) :=
{x ∈ Rn+1
∣∣∣∣ ∑i≤k≤j
xi ≥(j − i + 2
2
)}Loday, Realization of the Stasheff polytope (’04)
7654321
2
1
38
1
12
1123
321
213
312
141
LODAY’S ASSOCIAHEDRON
Asso(n) := conv {L(T) | T binary tree} = H ∩⋂
1≤i≤j≤n+1
H≥(i, j)
L(T) :=[`(T, i) · r(T, i)
]i∈[n+1]
H≥(i, j) :=
{x ∈ Rn+1
∣∣∣∣ ∑i≤k≤j
xi ≥(j − i + 2
2
)}Loday, Realization of the Stasheff polytope (’04)
7654321
2
1
38
1
12
1123
321
213 132
312 231
x1=x2 x2=x3
x1=x3 141
• Asso(n) obtained by deleting inequalities in the facet description of the permutahedron
• normal cone of L(T) in Asso(n) = {x ∈ H | xi < xj for all i→ j in T}=⋃σ∈L(T) normal cone of σ in Perm(n)
SPINES
spine of a tubing T = Hasse diagram of the inclusion poset of T
24 5
3
1 2 3
6
87 9
22
7
3
4
5
8
31
9
6
tube t of the tubing T 7−→ node s(t) of the spine S labeled
by t r⋃{t′ | t′ ∈ T, t′ ( t}
tube t(s) :=⋃{s′ | s′ ≤ s in S} ←− [ node s of the spine S
of the tubing T
S spine on G ⇐⇒ for each node s of S with children s1 . . . sk, the tubes t(s1) . . . t(sk)
lie in distinct connected components of G[t(s) r s
]
SPINES
spine of a tubing T = Hasse diagram of the inclusion poset of T
24 5
3
1 2 3
6
87 9
73
5
891
246
tube t of the tubing T 7−→ node s(t) of the spine S labeled
by t r⋃{t′ | t′ ∈ T, t′ ( t}
tube t(s) :=⋃{s′ | s′ ≤ s in S} ←− [ node s of the spine S
of the tubing T
S spine on G ⇐⇒ for each node s of S with children s1 . . . sk, the tubes t(s1) . . . t(sk)
lie in distinct connected components of G[t(s) r s
]
NESTED FANS AND GRAPH ASSOCIAHEDRA
THM. The collection of cones{{x ∈ H | xi < xj for all i→ j in T}
∣∣ T tubing on G}
forms a complete simplicial fan, called the nested fan of G. This fan is always polytopal.
Carr-Devadoss, Coxeter complexes and graph associahedra (’06)
Postnikov, Permutohedra, associahedra, and beyond (’09)
Zelevinsky, Nested complexes and their polyhedral realizations (’06)
4
2
3
1
4
2
1
34
2
31
3
2
4
1 (2,1,4,3)
(3,1,4,2)
(3,2,4,1)(4,1,2,3)
(4,2,3,1)
(4,1,2,3)
(1,2,3,4)(1,2,4,3)
(1,4,1,4)(3,1,2,4)
(2,1,3,4)
(3,2,1,4)
(4,2,1,3)
(4,4,1,1)
(1,7,1,1)
(1,4,4,1)
4
2
3
1 3
2
4
1
4
2
1
3
4
2
31
HOHLWEG-LANGE’S ASSOCIAHEDRA
11
22
312
776
54
3
21
for an arbitrary signature ε ∈ ±n+1,
Asso(ε) := conv {HL(T) | T ε-Cambrian tree}
with HL(T)j :=
{`(T, j) · r(T, j) if ε(j) = −
n + 2− `(T, j) · r(T, j) if ε(j) = +
Hohlweg-Lange, Realizations of the associahedron and cyclohedron (’07)
Lange-P., Using spines to revisit a construction of the associahedron (’15)
j
<j >j
?j
<j >j
?Rule for Cambrian trees
123
321
132
231
303
HOHLWEG-LANGE’S ASSOCIAHEDRA
11
22
312
776
54
3
21
for an arbitrary signature ε ∈ ±n+1,
Asso(ε) := conv {HL(T) | T ε-Cambrian tree}
with HL(T)j :=
{`(T, j) · r(T, j) if ε(j) = −
n + 2− `(T, j) · r(T, j) if ε(j) = +
Hohlweg-Lange, Realizations of the associahedron and cyclohedron (’07)
Lange-P., Using spines to revisit a construction of the associahedron (’15)
j
<j >j
?j
<j >j
?Rule for Cambrian trees
123
321
213 132
312 231
x1=x2 x2=x3
x1=x3303
• Asso(n) obtained by deleting inequalities in the facet description of the permutahedron
• normal cone of HL(T) in Asso(ε) = {x ∈ H | xi < xj for all i→ j in T}
SIGNED SPINES ON SIGNED TREES
T tree on the signed ground set V = V− t V+ (negative in white, positive in black)
Signed spine on T = directed and labeled tree S st
(i) the labels of the nodes of S form a partition of the signed ground set V
(ii) at a node of S labeled by U = U−tU+, the source label sets of the different incoming
arcs are subsets of distinct connected components of TrU−, while the sink label sets
of the different outgoing arcs are subsets of distinct connected components of TrU+
0
1
49
2 31
0
8 4 9
5
6 7
2
35678
2 7
0
1
49
368
5
SPINE COMPLEX
Signed spine complex S(T) = simplicial complex whose inclusion poset is isomorphic to
the poset of edge contractions on the signed spines of T
2
43
1
3
1
4
2
4
31
2
3
41
2
1
43
2
3
1
4
2
4
2
31
3
2
4
1
4
2
1
3
4
2
3
1
2
3
1
4
4
1
2
3
32
1423
14
132
412
34
42
13342
1
142
3 134
2134
2
132
4
342
1
142
3
12
34
24
13
123
4
124
3
234
1
234
1
123
4
124
3
SPINE FAN
For S spine on T, define C(S) := {x ∈ H | xu ≤ xv, for all arcs u→ v in S}
4
2
3
1
4
2
1
3
1
43
2
3
41
2
4
2
31
3
2
4
1 3
2
1
4
1
2
4
3
THEO. The collection of cones F(T) := {C(S) | S ∈ S(T)} defines a complete simplicial
fan on H, which we call the spine fan
SIGNED TREE ASSOCIAHEDRA
THM. The spine fan F(T) is the normal fan of the signed tree associahedron Asso(T),
defined equivalently as
(i) the convex hull of the points
a(S)v =
{∣∣ {π ∈ Π(S) | v ∈ π and rv /∈ π}∣∣ if v ∈ V−
|V| + 1−∣∣ {π ∈ Π(S) | v ∈ π and rv /∈ π}
∣∣ if v ∈ V+
for all maximal signed spines S ∈ S(T)
(ii) the intersection of the hyperplane H with the half-spaces
H≥(B) :=
{x ∈ RV
∣∣∣∣ ∑v∈B
xv ≥(|B| + 1
2
)}for all signed building blocks B ∈ B(T)
SIGNED TREE ASSOCIAHEDRA
The signed tree associahedron Asso(T) is sandwiched between the permutahedron Perm(V)
and the parallelepiped Para(T)∑u6=v∈V
[eu, ev] = Perm(T) ⊂ Asso(T) ⊂ Para(T) =∑u−v∈T
π(u − v) · [eu, ev]
WHAT SHOULD I TAKE HOMEFROM THIS TALK?
THREE FAMILIES OF REALIZATIONS
SECONDARY
POLYTOPE
Gelfand-Kapranov-Zelevinsky (’94)
Billera-Filliman-Sturmfels (’90)
LODAY’S
ASSOCIAHEDRON
Loday (’04)
Hohlweg-Lange (’07)
Hohlweg-Lange-Thomas (’12)
CHAP.-FOM.-ZEL.’S
ASSOCIAHEDRON
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p p p p p p p pp p p p p p p p
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pppppppppppppppppppppppppppppppppppppppppα2
α1+α2
α2+α3
α1+α2+α3
α1
α3
(Pictures by CFZ)
Chapoton-Fomin-Zelevinsky (’02)
Ceballos-Santos-Ziegler (’11)
TAKE HOME YOUR ASSOCIAHEDRA!
SECONDARY
POLYTOPEA
B
C
C
D
E
E
FF
G
D
G
B
H
K
L
L
M
K
M
N
N
OO
H
A
LODAY’S
ASSOCIAHEDRON
D
A
E
M
K
AB
B
C C
D
F
E
J
G
G
H
H
II
J
M
FL
L
K
CHAP.-FOM.-ZEL.’S
ASSOCIAHEDRON
A
E
B
C
C
D
D
E
A
F
J
G
H
H
I
I
J
K
K
LLMM
B
F
G
TAKE HOME YOUR ASSOCIAHEDRA!
SECONDARY
POLYTOPEA
B
C
C
D
E
E
FF
G
D
G
B
H
K
L
L
M
K
M
N
N
OO
H
A
LODAY’S
ASSOCIAHEDRON
D
A
E
M
K
AB
B
C C
D
F
E
J
G
G
H
H
II
J
M
FL
L
K
CHAP.-FOM.-ZEL.’S
ASSOCIAHEDRON
A
E
B
C
C
D
D
E
A
F
J
G
H
H
I
I
J
K
K
LLMM
B
F
G
Thibault Manneville & VP
Compatibility fans for graphical nested complexes
arXiv:1501.07152
VP
Signed tree associahedra
arXiv:1309.5222
THANK YOU
A
B
C
C
D
E
E F F
GD
G
B
H
K
L
L
M
K
M
N
N
OO
H
A
SECONDARY POLYTOPEGelfand-Kapranov-Zelevinsky (’94)
Billera-Filliman-Sturmfels (’90)
D
A
E
M K
A
B
B
C
C
D
FE
J
G
G
H
H
II
J
M
F
L
L K
LODAY’S ASSOCIAHEDRON
Loday (’04)
Hohlweg-Lange (’07)
Hohlweg-Lange-Thomas (’12)
A EB
CC
DD
E
A
F
JGH
H
I
I
JK
K
L
L
MM
B
FG
CHAPOTON-FOMIN-ZELEVINSKY’S ASSOCIAHEDRONChapoton-Fomin-Zelevinsky (’02)
Ceballos-Santos-Ziegler (’11)