Bi-cyclic 4-polytopes

Abstract

In a previous paper [2] we studied the facial structure of convex hulls of certain curves that lie on the torus$T^2 = \left\{ {(\cos 2\pi x, sin 2\pi x, cos 2\pi y, sin 2\pi y):\left| x \right| \leqq \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} , \left| y \right| \leqq \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} } \right\} \subseteq R^4 .$ In this paper we use the results of [2] to study structure of convex hulls of certain finite subsets ofT ². Specifically, we study the combinatorial structure of the polytopes whose vertex sets are finite subgroups ofT ². Such a subgroup may be represented by Λ/Z ², where Λ ⊇Z ² is some planar geometric lattice. We shall show how the facial structure of the polytope may be read directly off the lattice Λ. We call these polytopesbi-cyclic polytopes; a study of their properties is under preparation.