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,...,n_k) and defines a polytope as the convex hull of all x in the hyperplane x₁n₁+...+x_kn_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 n₁ ... n_k' where d=n₁+...+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 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, ..., 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.
  • 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.
  • 184,026 single WS
  • 16,040 CWS of the type 4+4 (non-trivial)
  • 1,122 CWS of the type 3+4 (non-trivial)
  • 6 CWS of the type 3+3 (trivial)
  • 36 CWS of the type 3+3+3 (non-trivial)
  • 116 CWS from combining the (C)WS for d=3 trivially with (1,1)
  • 308 of the above are r-minimal (C)WS
  • There are 25 further 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.