Balanced triangulations on few vertices and an implementation of cross-flips

A d-dimensional simplicial complex is balanced if the underlying graph is (d + 1)-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds without boundary. As a result we exhibit a vertexminimal balanced triangulation of the dunce hat and balanced triangulations of several surfaces and 3-manifolds on few vertices. In particular we obtain small balanced triangulations of the 3-sphere that are non-shellable or shellable but not vertex decomposable