Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that the images of $a$ and $b$ are disjoint. My question is:

Is there any necessary/sufficient condition on the knots $a$ and $b$ (or for the link $a\bigcup b$) for $\alpha$ and $\beta$ to commute in $\pi_1(M)$?

I believe that there are conclusions (what conclusions?) for satellite knots. Any comment or reference will be greatly appreciated.