Let $K$ be an oriented knot in $S^3$ together with a framing $n$. Let $K(a,b)$ be the oriented link obtained by taking $a$ copies of the $n$-pushoff of $K$ with the same same orientation as $K$ and $b$ copies of the $n$-pushoff of $K$ with the opposite orientation as $K$. In particular, $K(1,0) = K$ and $K(0,1)$ is the reverse of $K$.

What can be said about the signature function $\sigma_{K(a,b)}(\lambda)$ in terms of $\sigma_K(\lambda)$?

Edit: The answer is yes - in fact there is a nice formula for more general satellite constructions. In "Signatures of iterated torus knots" by Litherland, it is show that $\sigma_{K(a,b)}(\lambda) = \sigma_K(\lambda^{a-b})$ (See Theorem 2 and Remark 2 following it).

Can we say anything about the nullity $n_{K(a,b)}(\lambda)$ in terms of some info about $K$? In light of the above Edit, I should probably just ask about how the nullity function behaves under satellite operations.