Expand all Collapse all | Results 1 - 7 of 7 |
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 2011 (vol 64 pp. 102)
Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies We introduce a flow of a spatial graph and see how invariants for
spatial graphs and handlebody-links are derived from those for flowed
spatial graphs.
We define a new quandle (co)homology by introducing a subcomplex of the
rack chain complex.
Then we define quandle colorings and quandle cocycle invariants for
spatial graphs and handlebody-links.
Keywords:quandle cocycle invariant, knotted handlebody, spatial graph Categories:57M27, 57M15, 57M25 |
3. CJM 2008 (vol 60 pp. 1240)
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links We define a family of formal Khovanov brackets
of a colored link depending on two parameters.
The isomorphism classes of these brackets are
invariants of framed colored links.
The Bar-Natan functors applied to these brackets
produce Khovanov and Lee homology theories categorifying the colored
Jones polynomial. Further,
we study conditions under which
framed colored link cobordisms induce chain transformations between
our formal brackets. We conjecture that
for special choice of parameters, Khovanov and Lee homology theories
of colored links are functorial (up to sign).
Finally, we extend the Rasmussen invariant to links and give examples
where this invariant is a stronger obstruction to sliceness
than the multivariable Levine--Tristram signature.
Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism Categories:57M25, 57M27, 18G60 |
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 2004 (vol 56 pp. 1022)
Non-Orientable Surfaces and Dehn Surgeries Let $K$ be a knot in $S^3$. This paper is devoted to Dehn surgeries which create
$3$-manifolds containing a closed non-orientable surface $\ch S$. We look at the
slope ${p}/{q}$ of the surgery, the Euler characteristic $\chi(\ch S)$ of the
surface and the intersection number $s$ between $\ch S$ and the core of the Dehn
surgery. We prove that if $\chi(\hat S) \geq 15 - 3q$, then $s=1$. Furthermore,
if $s=1$ then $q\leq 4-3\chi(\ch S)$ or $K$ is cabled and $q\leq 8-5\chi(\ch S)$.
As consequence, if $K$ is hyperbolic and $\chi(\ch S)=-1$, then $q\leq 7$.
Keywords:Non-orientable surface, Dehn surgery, Intersection graphs Categories:57M25, 57N10, 57M15 |
6. CJM 2000 (vol 52 pp. 293)
Floer Homology for Knots and $\SU(2)$-Representations for Knot Complements and Cyclic Branched Covers |
Floer Homology for Knots and $\SU(2)$-Representations for Knot Complements and Cyclic Branched Covers In this article, using 3-orbifolds singular along a knot with
underlying space a homology sphere $Y^3$, the question of existence
of non-trivial and non-abelian $\SU(2)$-representations of the
fundamental group of cyclic branched covers of $Y^3$ along a knot
is studied. We first use Floer Homology for knots to derive an
existence result of non-abelian $\SU(2)$-representations of the
fundamental group of knot complements, for knots with a
non-vanishing equivariant signature. This provides information on
the existence of non-trivial and non-abelian
$\SU(2)$-representations of the fundamental group of cyclic
branched covers. We illustrate the method with some examples of
knots in $S^3$.
Categories:57R57, 57M12, 57M25, 57M05 |
7. CJM 1999 (vol 51 pp. 1035)
The Homology of Abelian Covers of Knotted Graphs Let $\tilde M$ be a regular branched cover of a homology 3-sphere
$M$ with deck group $G\cong \zt^d$ and branch set a trivalent graph
$\Gamma$; such a cover is determined by a coloring of the edges of
$\Gamma$ with elements of $G$. For each index-2 subgroup $H$ of
$G$, $M_H = \tilde M/H$ is a double branched cover of $M$. Sakuma
has proved that $H_1(\tilde M)$ is isomorphic, modulo 2-torsion, to
$\bigoplus_H H_1(M_H)$, and has shown that $H_1(\tilde M)$ is
determined up to isomorphism by $\bigoplus_H H_1(M_H)$ in certain
cases; specifically, when $d=2$ and the coloring is such that the
branch set of each cover $M_H\to M$ is connected, and when $d=3$
and $\Gamma$ is the complete graph $K_4$. We prove this for a
larger class of coverings: when $d=2$, for any coloring of a
connected graph; when $d=3$ or $4$, for an infinite class of
colored graphs; and when $d=5$, for a single coloring of the
Petersen graph.
Categories:57M12, 57M25, 57M15 |