Maybe the fact that the Jones polynomial is a well-defined invariant of knots qualifies. If you're willing to accept Reidemeister theorem, whose proof is itself somewhat easy to follow though more delicate than one might think, then using the skein relation this is really straightforward and can be explained to someone with no mathematical background and basically boils down to drawing pictures. So checking that it works is easy, but the fact that it works is fairly miraculous, and understanding this involves some deep mathematics.