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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 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 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 |
4. 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 |
5. CJM 1997 (vol 49 pp. 883)
Proof of a conjecture of Goulden and Jackson We prove an integration formula involving Jack polynomials
conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with
enumeration of maps in surfaces.
Categories:05E05, 43A85, 57M15 |