One tradition way to represent knots is via knot diagrams. For a discussion of knot diagrams, see "Knots, links, braids and 3-manifolds" by Prasolov and Sossinsky
The program SnapPea represents knots in the three-sphere by converting a given knot diagram into a triangulation of the knot's complement. The triangulation simplification techniques in SnapPea appear to recognize the unknot within a second, for all "human-sized" knot diagrams. For the software SnapPy (Python upgrade of SnapPea) and much else, see the website CompuTop, maintained by Nathan Dunfield.
Ivan Dynnikov in his paper "Three-page approach to knot theory" has given a new and very interesting data structure for representing knots. This also recognizes unknots very quickly, and has led to interesting developments in Heegaard Floer homology -- see the discussions there on grid-links.