Deciding contractibility of a non-simple curve on the boundary of a 3-manifold

Éric Colin de Verdière (1, 2) , Salman Parsa (3)

1 CNRS - Centre National de la Recherche Scientifique
2 LIGM - Laboratoire d'Informatique Gaspard-Monge
3 DI-ENS - Département d'informatique - ENS Paris

We present an algorithm for the following problem. Given a triangulated 3-manifold M and a (possibly non-simple) closed curve on the boundary of M, decide whether this curve is contractible in M. Our algorithm is combinatorial and runs in exponential time. This is the first algorithm that is specifically designed for this problem; its running time considerably improves upon the existing bounds implicit in the literature for the more general problem of contractibility of closed curves in a 3-manifold. The proof of the correctness of the algorithm relies on methods of 3-manifold topology and in particular on those used in the proof of the Loop Theorem.

Mots clés: 3-manifold; homotopy; algorithm; computational topology

Domaines: Géométrie algorithmique [cs.CG]; Complexité [cs.CC]; Topologie algébrique [math.AT]; Algorithme et structure de données [cs.DS]

Éric Colin de Verdière, Salman Parsa. Deciding contractibility of a non-simple curve on the boundary of a 3-manifold. Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 2017, Barcelone, Spain. pp.2691-2704, ⟨10.1137/1.9781611974782.178⟩. ⟨hal-01583536⟩