Unknotting numbers calculated according to the Conjecture coincide with all exactly determined unknotting numbers ( n £ 10) from the book A Survey of Knot Theory by A. Kawauchi (Appendix F) (1996) and Kawauchi's table updated with reference to recent unknotting number results (Livingston and Cha, 2004), if in all ambiguous cases (A=1 or 2; B=2 or 3) for the unknotting numbers we take A = 2, B = 3. Hence, if any of those unknotting numbers is smaller than its maximal estimated value, this will be a counterexample for the Conjecture. The Conjecture holds for all twocomponent links whose exact unlinking numbers were calculated by P.Kohn (1993). The complete list of estimated unknotting numbers for knots with n = 11 and n = 12 crossings computed by the authors according to Conjecture, using the program LinKnot, is included in the Table of knot invariants by C.Livingston and J.C.Cha. The Conjecture was first introduced by J.A. Bernhard in 1994 and then independently proposed by S. Jablan in 1995, when it was effectively used for the calculation of unknotting numbers of the knots with n £ 10 crossings. It is now included in the knot theory program LinKnot as the basis of the function UnR (webMathematica UnR) that calculates unknotting and unlinking numbers of rational KLs, and UnKnotLink (webMathematica UnKnotLink) that calculates unknotting (unlinking) number of any KL. Since the computation of signatures is included in these functions and used for the control of results, for a large number of KLs we can be sure that estimated unlinking numbers are the exact unlinking numbers. Y.Nakanishi (1996) and A.Stoimenow (2004) proved that the Conjecture holds for one subclass of rational KLs: all rational KLs with unlinking number one have an unlinking number one minimal diagram. If the Conjecture does not hold for all KLs, it may be true for some restricted class, e.g., for all rational KLs. In the worst case, even if it is not true in general, it gives the best possible upper bound for unlinking numbers.
