Linking numbers in rational homology <IMG BORDER="0" SRC="/tran/2004-356-09/S0002-9947-04-03423-3/gif-title0/img1.gif" >-spheres, cyclic branched covers and infinite cyclic covers期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Jó zef H Przytycki Akira Yasuhara

Institution:	Department of Mathematics, The George Washington University, Washington, DC 20052 ; Department of Mathematics, Tokyo Gakugei University, Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan

Abstract:	We study the linking numbers in a rational homology -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 -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 -sphere branched over a knot is a rational homology -sphere, the linking number of a pair in the preimage of a link in the -sphere is determined by the Goeritz/Seifert matrix of the knot.

Keywords:	Linking number rational homology $3$-sphere framed link covering space linking matrix Goeritz matrix Seifert matrix

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