Representing and recognizing 3-manifolds obtained from I-bundles over the Klein bottle

As it is well-known, the boundary of the orientable I-bundle $K X^sim I $ over the Klein bottle K is a torus; thus - in analogy with torus bundle construction (see [S]) - any integer matrix A of order two with determinant -1 (resp. +1) uniquely defines an orientable (resp. non-orientable) 3-manifold $(K X^sim I) cup (KX^sim I)/A$, which we denote by KB(A). In thepresent paper an algorithmic procedure is described, which allowsto construct, directly from any such matrix A, an edge-colouredgraph representing the manifold KB(A) associated to A. As a consequence, it is proved via regular genus (see [G]) thatthe Heegaard genus of any such manifold is less or equal to four;moreover, six elements of existing catalogues of orientable 3-manifolds represented by edge-coloured graphs (see [L] and [CC_2]) are directly recognized as manifolds of type KB(A).