We investigated systematically the knotting of nicked circular duplex DNA by Escherichia coli topoisomerase I. Agarose gel electrophoresis of knots forms a ladder of DNA bands. Each rung is made up of a variety of knots with the same number of nodes, or segment crossings; knots in adjacent rungs differ by one node. We extended the technique of electron microscopy of recA protein-coated DNA to the visualization of the complex knots tied by topoisomerase I. The striking result is that the enzyme produces every knot theoretically possible. The requirement for excess enzyme to form complex knots suggests a role for topoisomerase I in contorting the DNA in addition to promoting strand passage. We conclude that nodes formed are equally likely to be positive or negative and that topoisomerase I can pass DNA strands through a transient enzyme-generated break without regard to orientation of the passing strand. The results are interpreted in terms of a formulation for the topological requirements for knotting.