The basic idea of our classification scheme is to obtain all reflexive
polytopes of a given dimension as subpolytopes of certain maximal
The duals of these maximal polytopes are minimal polytopes.
Every maximal/minimal pair of polytopes can be represented with the help of a
(combined) weight system.
A weight system (WS) is a collection of positive integers
(n1,...,nk) and defines a polytope as the convex
hull of all x in the hyperplane
of Zk satisfying the inequalities xi >
A combined weight system (CWS) consists of several weight systems and
defines a collection of equations and a corresponding polytope.
A (C)WS is said to have the IP property if the corresponding polytope
has 0 in its interior. This is the case for every (C)WS we list here.
A polytope is called r-maximal if it is reflexive but not contained in
any other reflexive polytope.
A (C)WS is r-minimal if the corresponding polytope is r-maximal.
If a CWS has the IP property, then every WS belonging to it has the IP
Every reflexive polytope is a subpolytope of a polytope defined by a
(C)WS with the IP property, possibly on a sublattice.
For dimensions up to 4 every (C)WS with the IP property defines a
For dimensions up to 4 every reflexive polytope is a subpolytope
(again possibly on a sublattice) of a polytope defined by an r-minimal
In the files we present weight systems in the format
`d n1 ... nk' where
CWS are represented by several such entries in one line.
The type of a CWS is determined by the numbers of non-vanishing
weights for every single WS; we write k1+k2 for
a CWS containing a WS with k1 and another WS with
The weights are combined trivially if the CWS is of the form
(n11, ..., n1k1, 0, ..., 0;
0, ..., 0, n2,k1+1, ...,
and non-trivially if the positions of the non-zero entries in the
single WS overlap; similarly for more than two WS.
Polytopes are represented as
where #l and #c are the numbers of lines and columns of the matrix that
follows; this matrix represents the vertices of the polytope (either
as line or as column vectors)
For d=2 there are the three WS (1, 1, 1), (1, 1, 2), (1, 2, 3)
and the CWS (1, 1, 0, 0; 0, 0, 1, 1).
Only (1, 2, 3) is not r-minimal.
For d=3 there are 116 (C)WS and 16