This question is probably obvious to experts but I couldn't find the answer in the literature...
Background: Consider the mapping class group $Mod_g$ of the closed genus $g$ surface. There are many nice sets of generators (i.e. Humpreys famous $2g+1$ Dehn twists or Wajnryb's 2-element generating set etc).
If we consider the Torelli group $\mathcal{I}_g = Mod_g[1]$, we know it is finitely generated by bounding pair maps. D. Johnson proved that the next level in the Johnson filtration $Mod_g[2]$ (now called the Johnson kernel) is generated by Dehn twists around separating curves.
My question:
- Is there an explicit description of elements deeper in the mapping class group (i.e. in $Mod_g[k]$ with $k \geq 2$)? Or at least examples in picture form...
- Is there any known characterization (even for special values of $g$ and $k$) of such elements?
- Finally (unrealistically optimistic) - is there a known generating set?