Knot theory and complex curves in subcritical Stein domains

Beautiful results of Rudolph and Boileau-Orevkov characterize those links in the $3$-sphere which arise as transverse intersections with algebraic curves in $\mathbb{C}^2$.  I'll discuss work in progress which provides a corresponding characterization of such links in connected sums of $S^1\times  S^2$, viewed as the boundary of a subcritical Stein domain (the ball union Stein $1$-handles).  Time permitting, I'll talk about further generalizations being pursued with Baykur, Etnyre, Kawamuro, and Van Horn-Morris.