Topological Imitation of a Colored Link with the Same Dehn Surgery Manifold

By the topological imitation theory, we construct, from a given colored link, a new colored link with the same Dehn surgery manifold. In particular, we construct a link with a distinguished coloring whose Dehn surgery manifold is a given closed connected oriented 3-manifold except the 3-sphere. As a result, we can naturally generalize the difference between the Gordon-Luecke theorem and the property P conjecture to a difference between a link version of the Gordon-Luecke theorem and the Poincaré conjecture. Similarly, we construct a link with a -distinguished coloring whose Dehn surgery manifold is a given non-simply-connected closed connected oriented 3-manifold. We also construct a link with just two colorings whose Dehn surgery manifolds are the 3-sphere