Article electronically published on March 26, 2003 ON THE UNKNOTTING NUMBER OF MINIMAL DIAG.pdf

MATHEMATICS OF COMPUTATION Volume 72, Number 244, Pages 2043–2057 S 0025-5718(03)01514-X Article electronically published on March 26, 2003 ON THE UNKNOTTING NUMBER OF MINIMAL DIAGRAMS A. STOIMENOW Abstract. Answering negatively a question of Bleiler, we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent. 1. Introduction It is known that a diagram D of a knot can be made into a diagram of the unknot by crossing changes. The unknotting number u(D) of D is defined as the minimal number of such crossing changes, and the unknotting number u(K) of a knot K is given by u(K) = min D diagram of K u(D) . Taking the minimum over an infinite number of diagrams makes the unknotting number of a knot often hard to calculate (see [KM, Tr]). Thus one is led to consider modifications of this definition where only finitely many diagrams are considered. The most appealing idea is to consider just diagrams D of crossing number c(D) equal to the (minimal) crossing number c(K) for K. Thus define umin(K) = min D diagram of K with c(D) = c(K) u(D) and umax(K) = max D diagram of K with c(D) = c(K) u(D) . We have then for any knot K the obvious inequalities u(K) ≤ umin(K) ≤ umax(K) .(1) Suggestively, for many knots both inequalities are in fact equalities. The second inequality holds for alternating knots by the proof of the Tait crossing number [Ka, Mu, Th] and flyping conjectures [MT], and the first inequality is true for a large class of positive braid knots (including the torus knots) by the work of Murasugi [Mu2], Boileau-Weber [BW], and the proof of the (local) Thom conjecture [KMr]. It was surprising, when Bleiler [Bl] and Nakanishi [Na] (independently) found an example, 108 in the tables of [Ro, appendix], for which the first inequality was strict (u = 2, but umin = 3). Bleiler asked whether there are also examples of knots for which the second inequality is strict. This was confirmed b