Here is an example close to my heart. We show that the second coefficient of the Conway polynomial can be defined by counting quadrisecants. (Collinearities of $4$ points on the knot.)
(source: rybu.org)
Indeed, much of classical knot theory is done without resorting to planar diagram calculus. For example, the Alexander polynomial can be calculated from a Seifert surface for a knot, and most properties of the Alexander polynomial are best proved without trying to resort to diagrams (there are exceptions). Like Daniel Moskovich alludes to, the Alexander polynomial of a slice knot factorizes as $f(t)f(t^{-1})$, and the proof does not have anything to do with diagrams. More modern developments are also often diagram-free. Ozsvath and Szabo's Heegard-Floer homology was originally developed without diagrams. Only later was it figured out how to interpret the theory using planar diagrams, which was regarded as a good thing, since it made the problem of calculating it algorithmic, although incredibly computationally intensive.
