(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the four-ball genera of torus knots. As another illustration, we use calculate the invariant for several ten-crossing knots. 1.

Abstract. We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4-dimensional invariants. At least 10 % of the knots in Rolfsen’s table belong to this class of knots. We call them track knots. They are contained in the class of quasipositive knots. In this connection, we classify quasipositive knots and strongly quasipositive knots up to 10 crossings. 1.

. We show that a knot with a diagram with n granny and square tangles has unknotting number at least n, bridge number ? n, and braid index ? n. As an application, we construct exponentially many in the crossing number slice knots with arbitrarily high unknotting number. Keywords: tangle, Jones polynomial, unknotting number, braid index, bridge number, slice knot. AMS subject classification: 57M25 (primary), 57M15, 57N70 (secondary). 1. Introduction In a recent beautiful paper [Kr], David Krebes introduced an invariant of 4-tangles with values in Z \Theta Z=(p;q) (\Gamma p; \Gammaq) (a set one can think of as rational numbers with possibly zero denominators and not allowed to be reduced), that generalizes the classical iterated fraction of rational tangles, see [GK]. Up to a sign, p and q are the evaluations of the determinant D (\Gamma1) =V (\Gamma1) [J2, x12] on both closures of the tangle. Here D as the Alexander polynomial [Al] and V the Jones polynomial [J]. Krebes's invarian...

Abstract. 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

We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.