Suppose there are tow diagram $D_1$ and $D_2$ of knot $K$ with $c_1$ and $c_2$ crossing. Are there any bound of second type of Reidemeister move in term of $c_1$ and $c_2$? In other words, Are there Reidemeister moves like $\Omega_1,...,\Omega_n$ which transfer $D_1$ to $D_2$ and the number of second type of Reidemeister move in $\Omega_1,...,\Omega_n$ is bounded by $f(c_1,c_2)$? What is the best $f$?

I know Marc Lackenby found an upper bound for number of Reidemeister move but is there a better upper bound for the number of second type of Reidemeister move?