Asymptotics of the smooth $A_n$-realization problem

We solve an asymptotic variant of a smooth version of the $A_n$-realization problem for plane curves. As an application, we determine the cobordism distance between torus links of type $T(d,d)$ and $T(2,N)$ up to an error of at most $3d$. We also discuss the limits of knot theoretic approaches aimed at solving the $A_n$-realization problem.