A Partial Ordering of Knots Through Diagrammatic Unknotting

In this paper we define a partial order on the set of all knots and links using a special property derived from their minimal diagrams. A knot or link K′ is called a predecessor of a knot or link K if Cr(K′) < Cr(K) and a diagram of K′ can be obtained from a minimal diagram D of K by a single crossing change. In such a case we say that K′ < K. We investigate the sets of knots that can be obtained by single crossing changes over all minimal diagrams of a given knot. We show that these sets are specific for different knots and permit partial ordering of all the knots. Some interesting results are presented and many questions are posed.