I would like to show that 2 definitions of the linking number are equivalent.
Let $J$ and $K$ be compact, oriented differential manifolds of dimension 1 embedded in $\mathbb{R}^{3}$. Suppose $J \cap K =\emptyset$. Consider
$\lambda:J \times K \rightarrow S^{2}:(p,q)\mapsto \frac{q-p}{\lvert \lvert q-p \rvert \rvert}.$
Define $l_1(J,K):=\mathrm{deg}(\lambda)$.
The second definition is the following:
as defined in Rolfsens Knots and Links. Denote this number by $l_2(J,K)$. I want to show $l_1(J,K)=l_2(J,K)$.
I tried to follow the proof that Rolfsen gives in his book.
Let $z \in S^2$ be such that it lies above the projection plane and such that it corresponds with the viewers eye of the regular projection. I was able to show that $\lambda^{-1}(z)$ corresponds to crossings of $J$ under $K$. Now $z$ is regular hence deg$(\lambda)=\sum_{(p,q)\in\lambda^{-1}(z)}$ sgn$(\lambda,(p,q))$. Therefore we only need to show that sgn$(\lambda,(p,q))$ corresponds to the "correct" type of crossing in the regular projection for each $(p,q) \in \lambda^{-1}(z)$.
I don't know how to prove that these signs and crossings do indeed correspond. Does anyone know how to proceed? Thank you for your help!