Say we have a 2-component link $L$ with components $L_1$ and $L_2$. Are there known conditions that will ensure that there exists a Seifert surface $S$ of $L_1$ with curves $\alpha_1,\beta_1,...,\alpha_g,\beta_g$ representing a basis for $H_1(S)$ such that $lk(\alpha_i,L_2)=0=lk(\beta_i,L_2)$ for all $i=1,...,g$.
(eg. Vanishing linking number, Milnor invariants, etc. on $L$?)