In [this question][1] it is noted that $SL(2, \mathbb{R})/SL(2, \mathbb{Z})$ is homeomorphic to the trefoil complement in $S^3.$ Is there a similarly nice interpretation of $SL(2, \mathbb{R})/\Gamma(N)?$ for various $N?$


  [1]: http://mathoverflow.net/questions/93942/why-s3-k-and-sl2-r-sl2-z-are-diffeomorphic-here-k-is-a-trefoil-in-s3