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

Asymptotics of the smooth $A_n$-realization problem