Thistlethwaite's discovery of links with trivial Jones polynomial (MR1831681) should be mentioned here. He apparently discovered them "during the course of a routine computer ennumeration." Later he and Eliahou and Kauffman (MR1928648) were able to recognize these links as part of an infinite family of links with trivial Jones polynomial. Further, it's my impression that the calculation that these Jones polynomials are trivial doesn't use computers.