On unknotting numbers and four-dimensional clasp numbers of links

Abstract

Abstract. In this paper, we estimate the unknotting number and the fourdimensional clasp number of a link, considering the greatest euler characteristic for an oriented two-manifold in the four-ball bounded by the link. Combining with a result due to Rudolph, we prove that an inequality stronger than the Bennequin unknotting inequality actually holds for any link diagram. As an application we show the equality conjectured by Boileau and Weber for a closed positive braid diagram.

Unknotting sequences for torus knots

“…Concerning Lemma 2, we should mention that the equality u = g * holds for all positive braid knots. This follows easily from results of Kawamura [6] and Boileau-Weber [4].…”

Scissor equivalence for torus links

“…the minimal number of crossing changes needed to pass from one torus knot to another [5]. This may be more difficult since the cobordism distance of two knots provides a lower bound [6], but not an upper bound, for their Gordian distance. We do not even know if the correction term (corresponding to f in Theorem 2) is a sub-quadratic function for the Gordian distance of torus knots.…”

Knot concordances and alternating knots

“…In this section we will consider the count of double points in singular concordances between knots. In the case that one of the knots is the unknot, this is a well studied invariant; references include [3,13,23].…”