The Knife tool cuts across shapes, dividing them perfectly… but when you drag the knife, it makes a wavy, curvy line.
Yuck.
To make it cut straight, hold down ALT (PC) or Option (Mac).
Hold down the Shift key, too, if you want to cut perfect vertical, horizontal or 45 degree lines.
Wish I’d known this earlier! I was using the Pathfinder, so this tip saves me several steps each time I want to cut straight across an object.
I find that I am getting more useful information regarding Adobe products from sites like this, as opposed to the Adobe help sites themselves.