1999/07/14 by Marc Lackenby, Lackenby, Marc
Computer Science · Mathematics · #57M25 #57N10 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GT #msc:57M25 #msc:57N10
paper · pdf · doi:10.48550/arxiv.math/9907093
76 pages, 16 figures. For the same paper with better quality figures, visit http://www.dpmms.cam.ac.uk/~ml128
arxiv created 1999/07/14 · openalex publication_date 1999/07/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold MK(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of MK(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundary of M a (possibly empty) union of tori. Moreover, there is a simple algorithm to find all these surgery curves, which involves inserting tangles into the 3-simplices of any given triangulation of M. As a consequence, we deduce some results about the finiteness of certain unknotting operations on knots in the 3-sphere.