460 research outputs found
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
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