I've read that Elms are susceptible to summer branch drop, in which fairly large branches just drop off with no apparent cause, even in still, calm weather. I have observed this phenomenon in English walnut trees in southern CA. One time, a large 6" diameter branch dropped off and landed on the roof a my friend's car that was parked in the driveway. It broke off cleanly, right at the trunk of the tree. I was amazed the first time I saw this "branch drop" happen.
I don't know if that's what's happening to your tree, or not. Anthracnose and some other diseases can actually girdle branches and cause them to drop off. Since you describe a "cut" that "goes all the way around," you may want to research that.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams