The basic idea of our classification scheme is to obtain all reflexive
polytopes of a given dimension as subpolytopes of certain maximal
polytopes.
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
(n_{1},...,n_{k}) and defines a polytope as the convex
hull of all x in the hyperplane
x_{1}n_{1}+...+x_{k}n_{k}=0
of Z^{k} satisfying the inequalities x_{i} >
-2.
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.
Facts:
If a CWS has the IP property, then every WS belonging to it has the IP
property.
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
reflexive polytope.
For dimensions up to 4 every reflexive polytope is a subpolytope
(again possibly on a sublattice) of a polytope defined by an r-minimal
(C)WS.
Notation:
In the files we present weight systems in the format
`d n_{1} ... n_{k}' where
d=n_{1}+...+n_{k}.
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 k_{1}+k_{2} for
a CWS containing a WS with k_{1} and another WS with
k_{2} weights.
The weights are combined trivially if the CWS is of the form
(n_{11}, ..., n_{1k1}, 0, ..., 0;
0, ..., 0, n_{2,k1+1}, ...,
n_{2,k1+k2})
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
#l #c
line 1
...
line #l
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
r-maximal polytopes.
An outline of our classification scheme for reflexive polyhedra was
first presented in
hep-th/9512204.
The weight systems for d=3 and d=4 were computed in
alg-geom/9603007.
In that paper it was also shown that IP weights lead to reflexive
polytopes in up to 4 dimensions.