Abstract. Ozsváth and Szabó defined an analog of the Frøyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3-sphere branched over a knot K, we obtain an invariant δ of knot concordance. We show that δ is determined by the signature for alternating knots and knots with up to nine crossings, and conjecture a similar relation for all H-thin knots. We also use δ to prove that for all knots K with τ (K) > 0, the positive untwisted double of K is not smoothly slice.

Using the Heegaard Floer homology of Ozsváth and Szabó we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.In this section we study the relationship between a smooth four-manifold X and its boundary Y . We will assume throughout that X is negative definite. The following is an extension of [3, Lemma 3].Lemma 2.1 Let Y be a rational homology sphere; denote by h the order of H 1 (Y ; Z). Suppose that Y bounds X and denote by s the absolute value of the determinant of the intersection pairing on H 2 (X, Z)/Tors. Then h = st 2 , where st is the order of the image of H 2 (X; Z) in H 2 (Y ; Z), and t is the order of the image of the torsion subgroup of H 2 (X; Z).Proof. Note that for b 2 (X) > 0, X has a non-degenerate integer intersection formwe denote the absolute value of the determinant of this pairing by s. If b 2 (X) = 0, then set s = 1. The long exact sequence of the pair (X, Y ) yields the following (with integer coefficients):

We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsváth-Szabó's obstruction to unknotting number one. We determine the unknotting numbers of 9 10 , 9 13 , 9 35 , 9 38 , 10 53 , 10 101 and 10 120 ; this completes the table of unknotting numbers for prime knots with crossing number nine or less. Our obstruction uses a Kirby calculus description of a four-manifold W bounded by the branched double cover of the knot, and a theorem of Cochran and Lickorish which computes the signature of W .

Dedicated to José Maria Montesinos on the occasion of his 65th birthday.Abstract. Given a knot K in the three-sphere, we address the question: which Dehn surgeries on K bound negative-definite four-manifolds? We show that the answer depends on a number m(K), which is a smooth concordance invariant. We study the properties of this invariant, and compute it for torus knots.

Abstract. We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes required to obtain a slice knot; and the concordance unknotting number, which is the minimal unknotting number in a smooth concordance class. Using Heegaard Floer homology we obtain bounds that can be used to determine two of these invariants for all prime knots with crossing number ten or less, and to determine the concordance unknotting number for all but thirteen of these knots. As a further application we obtain some new bounds on Gordian distance between torus knots. We also give a strengthened version of Ozsváth and Szabó's obstruction to unknotting number one.

We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group L of links in S 3 , which has the concordance group of knots as a direct summand with infinitely generated complement. We consider variants of this using oriented and unoriented surfaces as well as smooth and locally flat embeddings.57M25, 57M27, 57N70

Abstract. The slicing number of a knot, u s (K), is the minimum number of crossing changes required to convert K to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus g s (K). We show that for many knots, previous bounds on unknotting number obtained by Ozsváth and Szabó and by the author in fact give bounds on the slicing number. Livingston defined another invariant U s (K) which takes into account signs of crossings changed to get a slice knot, and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots K n with slice genus n and Livingston invariant greater than n. Our bounds are based on restrictions (using Donaldson's diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.