3,021 research outputs found
Links with no exceptional surgeries
We show that if a knot admits a prime, twist-reduced diagram with at least 4
twist regions and at least 6 crossings per twist region, then every non-trivial
Dehn filling of that knot is hyperbolike. A similar statement holds for links.
We prove this using two arguments, one geometric and one combinatorial. The
combinatorial argument further implies that every link with at least 2 twist
regions and at least 6 crossings per twist region is hyperbolic and gives a
lower bound for the genus of a link.Comment: 28 pages, 15 figures. Minor rewording and organizational changes;
also added theorem giving a lower bound on the genus of these link
Explicit Dehn filling and Heegaard splittings
We prove an explicit, quantitative criterion that ensures the Heegaard
surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic
manifold X, and a Dehn filling whose meridian and longitude curves are longer
than 2pi(2g-1), we show that every genus g Heegaard splitting of the filled
manifold is isotopic to a splitting of the original manifold X. The analogous
statement holds for fillings of multiple boundary tori. This gives an effective
version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.Comment: 17 pages. v3 contains minor revisions and cleaner arguments,
incorporating referee comments. To appear in Communications in Analysis and
Geometr
Effective distance between nested Margulis tubes
We give sharp, effective bounds on the distance between tori of fixed
injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.Comment: 25 pages, 3 figures. v3 contains minor revisions. To appear in
Transactions of the AM
Cusp areas of Farey manifolds and applications to knot theory
This paper gives the first explicit, two-sided estimates on the cusp area of
once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link
complements. The input for these estimates is purely combinatorial data coming
from the Farey tesselation of the hyperbolic plane. The bounds on cusp area
lead to explicit bounds on the volume of Dehn fillings of these manifolds, for
example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the
Schreier normal form of the associated braid word. Finally, these results are
applied to derive relations between the Jones polynomial and the volume of
hyperbolic knots, and to disprove a related conjecture.Comment: 44 pages, 11 figures. Version 4 contains revisions and corrections
(most notably, in Sections 5 and 6) that incorporate referee comments. To
appear in the International Mathematics Research Notices
Hyperbolic semi-adequate links
We provide a diagrammatic criterion for semi-adequate links to be hyperbolic.
We also give a conjectural description of the satellite structures of
semi-adequate links. One application of our result is that the closures of
sufficiently complicated positive braids are hyperbolic links.Comment: 25 pages, 9 figure
Investigation of robotics-assisted tilt-table therapy for early-stage rehabilitation in spinal cord injury
This article provides the outcome of an investigation of robotics-assisted tilt-table therapy for early-stage rehabilitation in spinal cord injur
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