The Blanchfield pairing is usually defined on the homology of the infinite cyclic cover over the knot exterior.


In his article "cobordism of satellite knots", Litherland works with the $0$-framed surgery instead of the knot exterior. 


What is the difference between the two resulting pairings? 



Blanchfield pairing: knot exterior versus $0$-framed surgery
They are essentially identical.
Let $K$ be a knot with exterior $X$ and $M$ the closed 3-manifold
obtained by 0-framed (longitude) surgery on $K$, 
and let $X\_\infty$ and $M\_\infty$ be the infinite cyclic covers.
Thus $M\_\infty$ is obtained from $X\_\infty$ by adjoining a 2-cell to kill the longitude and then a 3-cell.
We may construct $X\_\infty$ by ``splitting" $X$ along a Seifert surface $S$.
Since $S$ has boundary a longitude and lifts to $X\_\infty$,
the longitude liftsc to $X\_\infty$, and each of its lifts is null-homologous there.
Hence the inclusion of $X\_\infty$ into $M\_\infty$
induces an isomorphism $H\_1(X\_\infty)\cong H\_1(M\_\infty)$,
which is easily seen to be an isometry of Blanchfield pairings.


