The second author and M. Sakuma gave a complete characterization of those essential simple loops on a 2-bridge sphere of a 2-bridge link which are nullhomotopic in the link complement. In this paper, we give an alternative proof to this result, by giving a simple proof for the small cancellation conditions of the upper presentations of 2-bridge link groups, which holds the key to the proof the result.
"A new proof for small cancellation conditions of 2-bridge link groups." Hiroshima Math. J. 42 (3) 411 - 423, November 2012. https://doi.org/10.32917/hmj/1355238375