Very briefly, the Conway notation for a knot is determined by first drawing circles in the plane, each meeting the knot in 4 points and so that they bound disjoint disks which together contain all knot crossing. The portion of the knot within each disk is a 2-stranded tangles. The selection of circles is done so that the tangles are each "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 which are not contained in any disk).
Conway's notation encodes the basic planar graphs and the tangles in very compact form. The reader is referred to Conway's original paper for details. For 11 crossing knots and less, we use Conway's original tabulation, along with corrections by Perko and others. For 12 crossing knots, the results were supplied by Slavik Jablan and
An enumeration of knots and links, and some of their algebraic properties, 1970 Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329--358 Pergamon, Oxford