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.)
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.
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.)
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.