Let $ P$ be a free abelian group of rank $ n-2$ with an integral symmetric bilinear form $ <,>$ . A sequence of elements $ A_1,\ldots,A_n$ in $ P$ is called an abstract toric system iff it satisfies the following conditions:

1) $ <A_i,A_{i+1}>=1$ for $ i\in[n]$

2) $ <A_i,A_j>=0$ for $ i\neq j$ and $ \{i,j\}\neq\{k,k+1\}$ and for all $ k\in[n]$

3) $ \sum_{i=1}^n<A_i,A_i>=12-3n$

where $ [n]$ means that the cyclic interval which means that we assume that the elements of $ [n]$ are in cyclic order in the sense that we consider $ [n]$ as a sysytem of representatives of $ \mathbb{Z}/n\mathbb{Z}$ .

Now, I assume that $ <A_i,A_i>\geq -2$ for each $ i$ . The question is for fixed $ n$ , say $ n=6$ , what is the classifications of sequences of $ (A_1^2,A_2^2,\ldots,A_6^2)$ , where $ A_i^2$ is short for $ <A_i,A_i>$ ?

This is actually what L.Hille and M.Perling called abstract toric system, they can associate a toric surface to this system and obtain the all the classification of sequence $ (A_1^2,\ldots,A_n^2)$ by classification of the corresponding toric surface. The question what I am asking is that Is there a purely combinatoric method to solve this problem without doing geometry as they did?