  This paper provides the complete table of prime knot projections with their
mirror images, without redundancy, up to eight double points systematically
thorough a finite procedure by flypes. In this paper, we show how to tabulate
the knot projections up to eight double points by listing tangles with at most
four double points by an approach with respect to rational tangles of J. H.
Conway. In other words, for a given prime knot projection of an alternating
knot, we show how to enumerate possible projections of the alternating knot.
Also to tabulate knot projections up to ambient isotopy, we introduce arrow
diagrams (oriented Gauss diagrams) of knot projections having no over/under
information of each crossing, which were originally introduced as arrow
diagrams of knot diagrams by M. Polyak and O. Viro. Each arrow diagram of a
knot projection completely detects the difference between the knot projection
and its mirror image.
