From colored triangulations to framed link presentations of 3-manifolds by a polynomial algorithm

Abstract

For a large class of closed orientable 3-manifolds, we present a polynomial algorithm to go from a triangulation to a framed link presenting the same manifold. The algorithm applies whenever we can get a resoluble gem from the triangulation. Most minimal gems have this property (about 95% of gems in our complete catalogue up to 28 vertices). From the gems which fail, nothing prevents another gem from inducing the same 3-manifold to be resoluble. We leave as an open problem presenting a 3-manifold which is not induced by a resoluble gem. This paper is an application of the theory of gems to 3-manifold topology.