Mutation classes of skew-symmetric 3x3-matrices

In this paper, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3x3-matrices and the set of triples of integers (a,b,c) which are all greater than 1 and where the product of the two smaller numbers is greater than or equal to the maximal number. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that these two cases are not intertwined.