Canadian Mathematical Society
www.cms.math.ca
|          |    |           |    |
|----------|----|-----------|----|
|          | |  |           | |  |
| Site map | |  | CMS store |    |

location:            Publications → journals → CJM
Abstract view

# The Homology of Abelian Covers of Knotted Graphs

Read article
[PDF: 356KB]
http://dx.doi.org/10.4153/CJM-1999-046-7
Canad. J. Math. 51(1999), 1035-1072
Published:1999-10-01
Printed: Oct 1999
• R. A. Litherland
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:

Format:  HTML  LaTeX  MathJax  PDF  PostScript

## Abstract

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 - 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]
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 24, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8545663356781006, "perplexity_flag": "middle"}