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The Homology of Abelian Covers of Knotted Graphs

Open Access article
 Printed: Oct 1999
  • R. A. Litherland
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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.
MSC Classifications: 57M12, 57M25, 57M15 show english descriptions Special coverings, e.g. branched
Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Relations with graph theory [See also 05Cxx]
57M12 - Special coverings, e.g. branched
57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45}
57M15 - Relations with graph theory [See also 05Cxx]

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