How do I cut a rope on a climb if I don't have a knife?
Falling with your rope running across a sharp edge is a proven method, but I suspect you are looking for a technique less final. I'm no Simon Yates, but I have learned to cut a rope, and without a knife, no less, usually in miserable stuck-rap-rope situations where you are forced to cut what rope you can salvage, and at miserable emergency retreats where you have to chop up your rope for anchor cord.
Know this: a rope under tension cuts as easily as flatulence on frijole night. To drum this point into me, a rope manufacturer once stretched a rope taut in a special rope-tensioning machine, then touched it with a dull plastic knife. The rope literally exploded into two bits. Eye-opening and sphincter-tightening. You, too, can use the wonders of rope mechanics to easily chop your cord without a knife. Here's how.
I call my first method the Hack Job. This requires a hammer, which you likely have if you are on a wall or putting up a route. First, pull the section of rope to be cut as tight as possible, and lay the rope against the rock. Hammer the spot you want to cut. It'll take 10 to 20 chafing blows to cut/beat the rope apart, but this really does the trick. No hammer? A fist-sized rock will suffice.
Alternately -- I call this the Alternate Method -- if you have a rack of pins, select a knife blade, pull the rope taut, place the blade against the rope and hammer on it. Quick and clean as a Bangkok rub-a-dub.
Last—the Last Method—if you don't have a hammer or a rock, pull the rope taut and saw it across a rock edge. Messy and slow, but less so than a rescue.