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1. CJM 2013 (vol 66 pp. 141)
Existence of Taut Foliations on Seifert Fibered Homology $3$-spheres This paper concerns the problem of existence of taut foliations among $3$-manifolds.
Since the contribution of David Gabai,
we know that closed $3$-manifolds with non-trivial second homology group
admit a taut foliation.
The essential part of this paper focuses on Seifert fibered homology $3$-spheres.
The result is quite different if they are integral or rational but non-integral homology $3$-spheres.
Concerning integral homology $3$-spheres, we can see that all but the $3$-sphere and the PoincarÃ© $3$-sphere admit a taut foliation.
Concerning non-integral homology $3$-spheres,
we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation.
Moreover, we show that the geometries do not determine the existence of taut foliations
on non-integral Seifert fibered homology $3$-spheres.
Keywords:homology 3-spheres, taut foliation, Seifert-fibered 3-manifolds Categories:57M25, 57M50, 57N10, 57M15 |
2. CJM 2010 (vol 62 pp. 994)
Curvature Bounds for Surfaces in Hyperbolic 3-Manifolds
A triangulation of a hyperbolic $3$-manifold is \emph{L-thick} if each
tetrahedron having all vertices in the thick part of $M$ is
$L$-bilipschitz diffeomorphic to the standard Euclidean tetrahedron.
We show that there exists a fixed constant $L$ such that every
complete hyperbolic $3$-manifold has an $L$-thick geodesic
triangulation. We use this to prove the existence of universal bounds on
the principal curvatures of $\pi_1$-injective surfaces and strongly
irreducible Heegaard surfaces in hyperbolic $3$-manifolds.
Category:57M50 |
3. CJM 2008 (vol 60 pp. 164)
Boundary Structure of Hyperbolic $3$-Manifolds Admitting Annular and Toroidal Fillings at Large Distance |
Boundary Structure of Hyperbolic $3$-Manifolds Admitting Annular and Toroidal Fillings at Large Distance For a hyperbolic $3$-manifold $M$ with a torus boundary component,
all but finitely many Dehn fillings yield hyperbolic $3$-manifolds.
In this paper, we will focus on the situation where
$M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling.
For such a situation, Gordon gave an upper bound of $5$ for the distance between such slopes.
Furthermore, the distance $4$ is realized only by two specific manifolds, and $5$
is realized by a single manifold.
These manifolds all have a union of two tori as their boundaries.
Also, there is a manifold with three tori as its boundary which realizes the distance $3$.
We show that if the distance is $3$ then the boundary of the manifold consists of at most three tori.
Keywords:Dehn filling, annular filling, toroidal filling, knot Categories:57M50, 57N10 |
4. CJM 2007 (vol 59 pp. 418)
On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$
contains Casson's knot invariant. Whether there are (essentially)
other Vassiliev knot invariants obtainable from $Q$ is an open
problem. We show that this is not so up to degree $9$. We also
give the (apparently) first examples of knots not distinguished
by 2-cable HOMFLY polynomials which are not mutants. Our calculations
provide evidence of a negative answer to the question whether Vassiliev
knot invariants of degree $d \le 10$ are determined by the HOMFLY and
Kauffman polynomials and their 2-cables, and for the existence of
algebras of such Vassiliev invariants not isomorphic to the algebras
of their weight systems.
Categories:57M25, 57M27, 20F36, 57M50 |
5. CJM 2006 (vol 58 pp. 673)
The Generalized Cuspidal Cohomology Problem Let $\Gamma \subset \SO(3,1)$ be a lattice.
The well known \emph{bending deformations}, introduced by
\linebreak Thurston
and Apanasov, can be used
to construct non-trivial curves of representations of $\Gamma$
into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains
an embedded totally geodesic surface. A tangent vector to such a
curve is given by a non-zero group cohomology class
in $\H^1(\Gamma, \mink{4})$. Our main result generalizes this
construction of cohomology to the context of ``branched''
totally geodesic surfaces.
We also consider a natural generalization of the famous
cuspidal cohomology problem for the Bianchi groups
(to coefficients in non-trivial representations), and
perform calculations in a finite range.
These calculations lead directly to an interesting example of a
link complement in $S^3$
which is not infinitesimally rigid in $\SO(4,1)$.
The first order deformations of this link complement are supported
on a piecewise totally geodesic $2$-complex.
Categories:57M50, 22E40 |