Trying to come up with a knight's move on my spherical chess

I thought a chess variant that takes place on a d60 would be cool, so I'm developing one. I can't come up with how the knight should move though, and I would like some help.  

As you should be able to see, each "triangle" of 3 cells has either: Maroon, Green, and Teal, or is all gray. Each green is surrounded by 2 grays, while Maroon and Teal are surrounded by 1. The bishops have been split into 3 pieces, one of which being the Jack. The bishops can either rotate around a gray triangle, or cross a diagonal, into a cell of the same color. It can choose to do this again without backtracking as long as the path is clear. The jack, however, will move into the second diagonal going clockwise around the square if it is gray, or it will move to the opposite diagonal according to the edges if the square is green. The rook moves how you'd expect- although this means it can go to the same cell 3 cells away 4 times over. The queen can do all of these 3 things. The most important thing I want the knight to do is to move to squares the queen can't reach. Additionally, I'd prefer it never move to a cell of the same color, but still move to "nearby" cells. Anyone want to help?