A favorite of mine is the chirality of the trefoil knot, which can be proved easily using the Jones polynomial or some of its relatives. Louis Kauffman's paper "New invariants in the theory of knots", http://homepages.math.uic.edu/~kauffman/Bracket.pdf explains this nicely.
I don't know how it was proved before the Jones polynomial, but quoting from p. 204 of Kauffman's paper, "In the old days (before 1984) this was something that required a lot of mathematical background."
