Splitting a Surface

I'm trying to split a surface of what is essentially an o-ring to show that the o-ring is in fact cut and then glued together.  Every time I try, the split line tool will only do it if the split covers two parts of the o-ring.  Does anyone know a cheat for this?  I tried 'single direction' and depending on the direction I choose, either I still get two splits or I get zero splits.  What I'm looking for is esentially the first image below.  What I get is either of the image below or a split line with no errors in the tree but no actual split.  This is wildly easy in Inventor, I feel like I must be missing something obvious or it just isn't possible in Solidworks.