On the Kinkiness of Closed Braids

In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel knots cannot be unknotted using only positive crossing changes. We also describe a subgroup of infinite rank in the smooth knot concordance group of which no element has a strongly quasipositive representative. In 1993, L. Rudolph proved a lower bound for the slice genus of a knot in terms of a presentation as the closure of a braid. It is clear that the same estimate holds for the number of double points of any properly immersed disk in the 4–ball spanning the knot, for such a disk which has r self–intersection points can be turned into an embedded surface of genus r by replacing all the self–intersection points by handles. In this paper, we show that there is a similar bound for the minimal number of positive self–intersection points of such an immersion, a knot invariant introduced by R. Gompf which is called the positive kinkiness. First let us recall the definition of the unknotting number and the positive respectively negative unknotting numbers of a link. Definition 1. Let K ⊂ S be an oriented link. 1. The unknotting number u(L) of L is the smallest number of crossing changes needed to alter L to the trivial link. 2. The positive (negative) unknotting number u+(L) (u−(L)) is the smallest number of positive (negative) crossing changes in any sequence of crossing changes deforming L into the trivial link. Here we say that a crossing change is positive if it replaces a positive crossing by a negative one, otherwise the crossing change is called negative. Note that positive and negative unknotting numbers may really depend on the orientation if the link has more than one component. If a link is described as the closure of a braid, we will always assume that the orientation is chosen such that all strings are oriented coherently, so that a generator produces a positive crossing. In [6], R. Gompf introduced the notion of the kinkiness of a knot. We restate his definition in a slightly modified form to include the case of a link with more than one component. In the sequel, we will assume that all immersions of surfaces in the 4–ball are smooth and proper in the sense that the only singularities are transverse double points and that they are embeddings near the boundary. Definition 2. Let L ⊂ S be an oriented link. The positive (negative) kinkiness κ+(L) (κ−(L)) of L is the smallest number of positive (negative) double points of a proper immersion F →֒ D with ∂F = L, where F is a connected surface of genus 0. The kinkiness of an oriented link L is the pair (κ+(L), κ−(L)). Date: February 1, 2008.