Recently I read Dale Rolfsen's paper –A surgical view of Alexander’s polynomial. This is a good paper. But there is a lemma which I don’t know how to prove.The lemma is following:


Lemma:In the cover$R^{1}\times R^{2}\to S^{3}$- {trivial knot},let $\alpha,\beta$ be disjoint closed oriented curves downstairs which lift to closed oriented curves $\tilde{\alpha},\tilde{\beta}$ in $R^{1}\times R^{2}$. Then the following formula holds:


$lk(\alpha,\beta)=\sum­\_{k}lk(\tilde{\alpha},t^{k}\tilde{\beta})$.


assuming the covering map preserves orientation.


Thanks a lot!



linking number and covering
The kind of argument you'd see Rolfsen make in his textbook would be to consider how one constructs the abelian cover explicitly via the Seifert surface of the trivial knot. This is a disc. Ensure it intersects your curves transversely. This allows you to write a simplified diagram for your curves, consisting of a tangle + some unknotted, unlinked semi-circles that crash through the disc. 


From there, you can construct diagrams for the lift to the covering space quite explicitly and the crossing-number formula is immediate. 


