12,691 research outputs found
Geodesics and compression bodies
We consider hyperbolic structures on the compression body C with genus 2
positive boundary and genus 1 negative boundary. Note that C deformation
retracts to the union of the torus boundary and a single arc with its endpoints
on the torus. We call this arc the core tunnel of C. We conjecture that, in any
geometrically finite structure on C, the core tunnel is isotopic to a geodesic.
By considering Ford domains, we show this conjecture holds for many
geometrically finite structures. Additionally, we give an algorithm to compute
the Ford domain of such a manifold, and a procedure which has been implemented
to visualize many of these Ford domains. Our computer implementation gives
further evidence for the conjecture.Comment: 31 pages, 11 figures. V2 contains minor changes. To appear in
Experimental Mathematic
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
Treewidth, crushing, and hyperbolic volume
We prove that there exists a universal constant such that any closed
hyperbolic 3-manifold admits a triangulation of treewidth at most times its
volume. The converse is not true: we show there exists a sequence of hyperbolic
3-manifolds of bounded treewidth but volume approaching infinity. Along the
way, we prove that crushing a normal surface in a triangulation does not
increase the carving-width, and hence crushing any number of normal surfaces in
a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former
argument (in V1) used a construction that relied on a wrong theorem. Section
5.1 has also been adjusted to the new construction. Various other arguments
have been clarifie
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
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