The Conway notation for a knot is determined by first drawing circles in the plane, each of which meets the knot in 4 points and bounds a disjoint disk. The disks together contain all knot crossings. The portion of the knot within each disk is a 2-stranded tangle. The circles are chosen so that the tangles are all "rational tangles," which can be described with a sequence of integers. The circles in the plane determine a planar graph, each circle representing a vertex, each with valence four (the edges arising from the strands of the knot that are not contained in any disk).
Conway's notation encodes the basic planar graphs and the tangles in a very compact form. See [1] for details. For 11 and fewer crossing knots, we use Conway's original tabulation, along with corrections by Perko and others. Results for 12 crossing knots were supplied by Slavik Jablan and Radmila Sazdanovic.
[1] Conway, J. H., "An enumeration of knots and links, and some of their algebraic properties," Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford (1970), 329-358.