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

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.