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Linking numbers in rational homology $3$-spheres, cyclic branched covers and infinite cyclic covers


Authors: Józef H. Przytycki and Akira Yasuhara
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3669-3685
MSC (2000): Primary 57M25; Secondary 57M10, 57M12
DOI: https://doi.org/10.1090/S0002-9947-04-03423-3
Published electronically: January 16, 2004
MathSciNet review: 2055749
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Abstract: We study the linking numbers in a rational homology $3$-sphere and in the infinite cyclic cover of the complement of a knot. They take values in $\mathbb{Q}$ and in ${Q}(\mathbb{Z}[t,t^{-1}])$, respectively, where ${Q}(\mathbb{Z}[t,t^{-1}])$ denotes the quotient field of $\mathbb{Z}[t,t^{-1}]$. It is known that the modulo- $\mathbb{Z}$ linking number in the rational homology $3$-sphere is determined by the linking matrix of the framed link and that the modulo- $\mathbb{Z}[t,t^{-1}]$ linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate `modulo  $\mathbb{Z}$' and `modulo  $\mathbb{Z}[t,t^{-1}]$'. When the finite cyclic cover of the $3$-sphere branched over a knot is a rational homology $3$-sphere, the linking number of a pair in the preimage of a link in the $3$-sphere is determined by the Goeritz/Seifert matrix of the knot.


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Additional Information

Józef H. Przytycki
Affiliation: Department of Mathematics, The George Washington University, Washington, DC 20052
Email: przytyck@research.circ.gwu.edu

Akira Yasuhara
Affiliation: Department of Mathematics, Tokyo Gakugei University, Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan
Email: yasuhara@u-gakugei.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-04-03423-3
Keywords: Linking number, rational homology $3$-sphere, framed link, covering space, linking matrix, Goeritz matrix, Seifert matrix
Received by editor(s): December 1, 2001
Received by editor(s) in revised form: May 1, 2003
Published electronically: January 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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