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1. 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 |
2. 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 |