The main theorem of [this paper of Scharlemann][1] implies that a split two component link which is not the unlink cannot have a band sum giving the unknot. 
See also [Eudave-Muñoz][2] and [Ishihara-Motegi][3] and [Bao][4]. This move is called "band surgery" or "H(2) move" (after [Hoste-Nakanishi-Taniyama][5]).


  [1]: http://link.springer.com/article/10.1007%2FBF01388659
  [2]: http://www.jstor.org/stable/2153918
  [3]: http://www.ams.org/mathscinet-getitem?mr=2757689
  [4]: http://dx.doi.org/10.1016/j.topol.2012.02.008
  [5]: http://www.ams.org/mathscinet-getitem?mr=1075165