Is there a name for incompletely-sliced potatoes?
Potatoes that have been sliced almost all the way, buttered, spiced and then roasted until they fan out often go by the name Hasselback Potatoes. 

Reputedly they were invented by a Swedish chef in the 1950s and the name is based on the name of the Swedish restaurant, Hasselbacken, in Stockholm.

I don't think there is a specific culinary term for "slicing almost all the way".