## Classification scheme and weight systems

• 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 (n1,...,nk) and defines a polytope as the convex hull of all x in the hyperplane x1n1+...+xknk=0 of Zk satisfying the inequalities xi > -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 n1 ... nk' where d=n1+...+nk.
• 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 k2 weights.
• The weights are combined trivially if the CWS is of the form
(n11, ..., n1k1, 0, ..., 0; 0, ..., 0, n2,k1+1, ..., n2,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.
• 95 single WS
• 17 CWS of the type 3+3 (non-trivial)
• The 4 CWS from combining the (C)WS for d=2 trivially with (1,1)
• 15 of the above are r-minimal (C)WS
• The 16'th r-maximal polytope is given by
```   3 4
1 1 1 -3
0 2 0 -2
0 0 4 -4
```
• For d=4 there are the following 201,346 (C)WS and 333 r-maximal polytopes.
• Bibliography:
• 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.
• The CWS were worked out in hep-th/9703003.
• The full classification was achieved in hep-th/9805190 for d=3 and in hep-th/0002240 for d=4.
• A mathematical description of the algorithm in its final form can be found in math.AG/0001106.