Dehn filling of the "magic" 3-manifold

We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: - for every integer n we can prove that there are infinitely many hyperbolic knots in the 3-sphere having exceptional surgeries n, n+1, n+2, n+3, with n+1, n+2 giving small Seifert manifolds and n, n+3 giving toroidal manifolds; - we exhibit a 2-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements; - we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements; - we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling.Comment: 56 pages, 10 figures, 16 tables. Some consequences of the classification adde

Algorithmic simplification of knot diagrams: new moves and experiments

This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diagram D' with the minimal possible number of crossings for the isotopy class of the knot represented by D. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain Z1, Z2, Z3 generalizing the classical Reidemeister moves R1, R2, R3, and another one C (together with a variant) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones. We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web.Comment: 38 pages, 33 figure

Two-sided bounds for the complexity of cyclic branched coverings of two-bridge links

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate uses the hyperbolic volume and results of Cao-Meyerhoff and Gueritaud-Futer (who recently improved previous work of Lackenby), while the upper estimate is based on an explicit triangulation, which also allows us to give a bound on the Delzant T-invariant of the fundamental group of the manifold.Comment: Estimates improved using recent results of Gueritaud-Futer and Kim-Ki