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

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

2001-11-19

arxiv.org/abs/math/0111203v1

Mathematics

Geometric Topology

LaTeX, 24 pages, 6 figures

Scientific paper

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 $\Bbb Q$ and in ${Q}({\Bbb Z}[t,t^{-1}])$ respectively, where ${Q}({\Bbb Z}[t,t^{-1}])$ denotes the quotient field of ${\Bbb Z}[t,t^{-1}]$. It is known that the modulo-$\Bbb Z$ linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-${\Bbb Z}[t,t^{-1}]$ 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 $\Bbb Z$' and ` modulo ${\Bbb Z}[t,t^{-1}]$'. 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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