  We derive the exact critical line of the O($n$) loop model on the martini
lattice as a function of the loop weight $n$.A finite-size scaling analysis
based on transfer matrix calculations is also performed.The numerical results
coincide with the theoretical predictions with an accuracy up to 9 decimal
places. In the limit $n\to 0$, this gives the exact connective constant
$\mu=1.7505645579...$ of self-avoiding walks on the martini lattice. Using
similar numerical methods, we also study the O($n$) loop model on the 3-12
lattice. We obtain similarly precise agreement with the exact critical points
given by Batchelor [J. Stat. Phys. 92, 1203 (1998)].
