A Mysterious Duality Relation for 4-dimensional Polytopes.

Combinatorics and more 2018-08-02

Two dimensions

Before we talk about 4 dimensions let us recall some basic facts about 2 dimensions:

A planar polygon has the same number of vertices and edges.

This fact, which just asserts that the Euler characteristic of $S^1$ is zero, can be reformulated as: Polygons and their duals have the same number of vertices.

A linear algebra statement

The spaces of affine dependencies of the vertices for a polygon P and its dual P* have the same dimension.

I am not aware of a pairing or an isomorphism for demonstrating this equality. (See, however Tom Braden’s comment.)

Four dimensions

Given a 4-dimensional polytope P define $\gamma (P) =f_1(P) + f_{02}(P)-3f_2(P)-4f_0+10.$

Here, $f_i(P)$ is the number of $i$ -faces of $P$ . $f_{02}(P)$ is the number of chains of the form 0-face $\subset$ 2-face. In other words it is the sum over all 2-faces of P, of the number of their vertices. In 1987 I discovered the following:

Theorem [Mysterious Four-dimensional Duality]: Let P and P* be dual four dimensional polytopes then

(1) γ(P)=γ(P*)

Here is an example: Let $P$ be the four dimensional cube. In this case $f_0=16$ , $f_1=32$ , $f_2=24$ , $f_3=8$ , and since every 2-face has 4 vertices $f_{02}=96$ . $\gamma (P)=32+96-72-80+10=2$ . $P^*$ is the 4-dimensional cross-polytope with $f_0=8$ , $f_1=24$ , $f_2=32$ , $f_3=16$ , and since every 2-face has 3 vertices $f_{02}=96$ . $\gamma (P^*)=24+96-96-40+10=2$ . Viola!

The proof of (1) is quite a simple consequence of Euler’s theorem.

For the 120-cell and its dual the 600-cell $\gamma=250$ .

Frameworks, rigidity and Alexandrov-Whiteley’s theorem.

A framework based on a $d$ -polytope $P$ is obtained from $P$ by adding diagonals which triangulate every 2-face of $P$ . Adding the diagonals leads to an infinitesimally rigid framework. This was proved by Alexandrov for 3 dimension and by Walter Whiteley in higher dimensions.

Walter Whiteley

Theorem (Alexandrov-Whiteley): For $d\ge 3$ , every famework based on $P$ is infinitesimmaly rigid.

It follows from the Alexandrov-Whiteley theorem that for a four-dimensional polytope $P$ , $\gamma (P)$ is the dimension of the space of stresses of every framework based on $P$ . (An affine stress is an assignment of weights to edges so that every vertex lies in “equilibrium”.) Whiteley’s theorem implies that $\gamma (P) \ge 0$ for every 4-dimensional polytope $P$ .

4-dimensional stress-duality: Let P and P* be dual four dimensional polytopes. Then the space of affine stresses for a frameworks based on P and on P* have the same dimension.

Again, I am not aware of a pairing or an isomorphism that demonstrates this equality between dimensions. (See, however Tom Braden’s comment.)

Remark: Given a $d-$ polytope $P$ , let $f_1^+(P)$ be the number of edges in a framework based on $P$ . We can define for every $d$ -polytope, $\gamma (P)=f_1^+(P)-df_0(P)+{{d+1} \choose {2}}$ . Whiteley’s theorem implies that $\gamma (P) \ge 0$ for every $P$ . (For $d=3$ by Euler’s theorem $\gamma =0$ .)

Just as a polytope in dimension greater than 2 need not have the same number of vertices as its dual, it is also no longer true in dimensions greater than 4 that $\gamma(P)$ equals $\gamma (P^*)$ . However, it is true in every dimension that $\gamma (P)=0$ if and only if $\gamma (P^*)=0$ .

Toric varieties

Let $T(P)$ be a toric variety based on a rational 4-dimensional polytope $P$ . The dimension of the primitive part of the 4th intersection homology group of $T(P)$ is equal to $\gamma (P)$ . Our duality theorem thus asserts that the primitive part of the 4th intersection homology group of $T(P)$ has the same dimension as the primitive part of the 4th intersection homology group of $T(P^*)$ .

Some references: Whiteley’s theorem (Whiteley, 1984); The 4-d duality relation (Kalai, 1987); A more general relation (Bayer and Klapper, 1991); An even more general relation (Stanley; 1992) Related theorems on combinatorics of polytopes (Braden, 2006); Connection to Mirror symmetry (Batyrev and Borisov, 1995); Connection to Koszul duality (Braden, 2007); Rigidity and polarity in 3-d (Whiteley 1987)