  Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of
crossing exchanges and smoothings on $D$ that yields a diagram of $L$? We
approach this problem from the computational complexity point of view. It
follows from work by Endo, Itoh, and Taniyama that if $L$ is a prime link with
crossing number at most $5$, then there is an algorithm that answers this
question in polynomial time. We show that the same holds for all torus links
$T_{2,m}$ and all twist knots.
