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Edge-Transitive Polytopes
Professorship for Algorithmic and Discrete Mathematics
Edge-Transitive Polytopes
Martin Winter
Professorship for Algorithmic and Discrete Mathematics
08. November, 2019
DiscMath · 08. November, 2019 · Martin Winter 1 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Symmetries of Polytopes
Regular polytopes (classification: Schlafli, 1852)
dim 2 3 4 5 6 7 8 9 · · ·# ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s
Definition.I regular := flag-transitive
flag := (vertex ⊂ edge ⊂ face).
DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Regular polytopes (classification: Schlafli, 1852)
dim 2 3 4 5 6 7 8 9 · · ·# ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s
Definition.I regular := flag-transitive
flag := (vertex ⊂ edge ⊂ face).
DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Regular polytopes (classification: Schlafli, 1852)
dim 2 3 4 5 6 7 8 9 · · ·# ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s
Definition.I regular := flag-transitive
flag := (vertex ⊂ edge ⊂ · · · ⊂ facet).
DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Vertex-transitive polytopes
Theorem. (Babai, 1977; Ladisch, 2014)
Almost every finite group is the symmetry group of a vertex-transitive polytope.
Examples: Birkhoff polytope, TSP polytopes, ...
Keywords: orbit polytopes, representation polytopes, ...
DiscMath · 08. November, 2019 · Martin Winter 3 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Vertex-transitive polytopes
Theorem. (Babai, 1977; Ladisch, 2014)
Almost every finite group is the symmetry group of a vertex-transitive polytope.
Examples: Birkhoff polytope, TSP polytopes, ...
Keywords: orbit polytopes, representation polytopes, ...
DiscMath · 08. November, 2019 · Martin Winter 3 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Vertex-transitive polytopes
Theorem. (Babai, 1977; Ladisch, 2014)
Almost every finite group is the symmetry group of a vertex-transitive polytope.
Examples: Birkhoff polytope, TSP polytopes, ...
Keywords: orbit polytopes, representation polytopes, ...
DiscMath · 08. November, 2019 · Martin Winter 3 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Edge-transitive polytopes
Edge-transitivity in R3
Theorem. (Grunbaum & Shephard, 1987)
There are nine edge-transitive polyhedra.
DiscMath · 08. November, 2019 · Martin Winter 4 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Starting a classification ...
Question.
Are there half-transitive polytopes?
What about half-transitive abstract polytopes?
DiscMath · 08. November, 2019 · Martin Winter 5 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Just edge-transitive polytopes
rhombic dodecahedron rhombic triacontahedron
Question.
Are there just edge-transitive polytopes in d ≥ 4 dimensions?
DiscMath · 08. November, 2019 · Martin Winter 6 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Just edge-transitive polytopes
rhombic dodecahedron rhombic triacontahedron
Question.
Are there just edge-transitive polytopes in d ≥ 4 dimensions?
DiscMath · 08. November, 2019 · Martin Winter 6 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Just edge-transitive polytopes
Some thoughts:
I Edge graph must be bipartite =⇒ 2-faces are 2n-gons.
I Zonotopes might be a good place to start looking.
Theorem. (W., 2019+)
There are no just edge-transitive zonotopes in ≥ 4 dimensions.
DiscMath · 08. November, 2019 · Martin Winter 7 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Just edge-transitive polytopes
Some thoughts:
I Edge graph must be bipartite =⇒ 2-faces are 2n-gons.
I Zonotopes might be a good place to start looking.
Theorem. (W., 2019+)
There are no just edge-transitive zonotopes in ≥ 4 dimensions.
DiscMath · 08. November, 2019 · Martin Winter 7 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
Arc-transitive polytopes
Arc-transitivity in R3
There are seven arc-transitive polyhedra:
DiscMath · 08. November, 2019 · Martin Winter 8 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
Arc-transitivity in R4
There are 15 known edge-transitive 4-polytopes:
I six regular 4-polytopes→ 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,
I five rectifications→ of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),
I two bitruncations→ of 4-simplex and 24-cell,
I two runcinations→ of 4-simplex and 24-cell,
+ an infinite family of (p, p)-duoprisms.
All of these are uniform polytopes. (in fact, Wythoffian)
DiscMath · 08. November, 2019 · Martin Winter 9 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
Arc-transitivity in R4
There are 15 known edge-transitive 4-polytopes:
I six regular 4-polytopes→ 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,
I five rectifications→ of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),
I two bitruncations→ of 4-simplex and 24-cell,
I two runcinations→ of 4-simplex and 24-cell,
+ an infinite family of (p, p)-duoprisms.
All of these are uniform polytopes.
(in fact, Wythoffian)
DiscMath · 08. November, 2019 · Martin Winter 9 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
Arc-transitivity in R4
There are 15 known edge-transitive 4-polytopes:
I six regular 4-polytopes→ 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,
I five rectifications→ of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),
I two bitruncations→ of 4-simplex and 24-cell,
I two runcinations→ of 4-simplex and 24-cell,
+ an infinite family of (p, p)-duoprisms.
All of these are uniform polytopes. (in fact, Wythoffian)
DiscMath · 08. November, 2019 · Martin Winter 9 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
DiscMath · 08. November, 2019 · Martin Winter 10 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
DiscMath · 08. November, 2019 · Martin Winter 11 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
DiscMath · 08. November, 2019 · Martin Winter 12 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
Number of Wythoffian arc-transitive polytopes
dim 1 2 3 4 5 6 7 8 9 10 11 12 13 ...
irred. 1 0 7 15 11 19 22 25 19 21 23 25 27 ...
prod. 0 0 0 0 0 6 0 14 6 10 0 38 0 ...
prisms 0 ∞ 0 ∞ 0 ∞ 0 ∞ 0 ∞ 0 ∞ 0 ...∑1
∞0 7
∞15 11
∞25 22
∞39 25
∞31 23
∞63 27 ...
#irred.(n) = 2n+ 1, for n ≥ 9.
DiscMath · 08. November, 2019 · Martin Winter 13 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
Non-Wythoffian arc-transitive polytopes?
Are these lists complete?
Question.
Are there non-uniform arc-transitive polytopes.
... or stronger ...
Question.
Are there non-Wythoffian arc-transitive polytopes.
DiscMath · 08. November, 2019 · Martin Winter 14 / 20 www.tu-chemnitz.de
Spectral methods
Spectral methods
Eigenpolytopes
G =⇒
0 1 0 · · ·1 00
. . ....
=⇒ θ1 ≥ θ2
↑u1, ..., ud ∈ Rn
≥ · · · ≥ θm =⇒
| |u1 · · · ud| |
DiscMath · 08. November, 2019 · Martin Winter 15 / 20 www.tu-chemnitz.de
Spectral methods
Arc-transitive polytopes as eigenpolytopes
Conjecture.
An arc-transitive polytope P is the θ2-eigenpolytope of its edge-graph.
Consequences:
I P is uniquely determined by its edge-graph.
I P realizes all the symmetries of its edge-graph.
I For edge-length ` and circumradius r holds
r
`=
√deg(GP )
2λ2(L)L ... Laplacian.
I P is a perfect polytope.
I Every projection of P is either not arc-transitive, or has a different edge-graph.
DiscMath · 08. November, 2019 · Martin Winter 16 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
Distance-transitive polytopes
Distance-transitive graphs
Definition. Distance-transitivity
A graph G is distance-transitive, if for any i, j, ı, ∈ V (G) with dist(i, j) = dist(ı, )there is a φ ∈ Aut(G) with φ(i) = ı and φ(j) = .
DiscMath · 08. November, 2019 · Martin Winter 17 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
Distance-transitive graphs
DiscMath · 08. November, 2019 · Martin Winter 18 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
Distance-transitive polytopes
Theorem.A distance-transitive polytope P is the θ2-eigenpolytope of its edge-graph.
Consequences:
Theorem.If P ⊂ Rd is a distance-transitive polytope, then
I P is the unique distance-transitive polytope with this edge-graph.
I P realizes all the symmetries of its edge-graph.
DiscMath · 08. November, 2019 · Martin Winter 19 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
Classification of distance-transitive polytopes
Theorem. (Godsil, 1997; W., 2019+)
If P ⊂ Rd is a distance-transitive polytope, then P is one of the following:
I a regular polygon,
I the icosahedron,
I the dodecahedron,
I a cross-polytope,
I a hyper-simplex (this includes regular simplices),
I a demi-cube,
I a cartesian power of a simplex (this includes hypercubes),
I the 6-dimensional 221-polytope,
I the 7-dimensional 321-polytope.
DiscMath · 08. November, 2019 · Martin Winter 20 / 20 www.tu-chemnitz.de
The EndQuestions?