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?