Related papers: Computations of the Ozsvath-Szabo knot concordance…

Based on work of Rasmussen, we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau invariant.

The slice-Bennequin inequality states an upper bound for the self-linking number of a knot in terms of its four-ball genus. The $s$-Bennequin and $\tau$-Bennequin inequalities provide upper bounds on the self-linking number of a knot in…

Let \nu be any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, |\nu(K)| <= g_4(K), and determines the 4-ball genus of positive torus knots, \nu(T_{p,q}) = (p-1)(q-1)/2. Either of the knot concordance…

As proved by Hedden and Ording, there exist knots for which the Ozsvath-Szabo and Rasmussen smooth concordance invariants, tau and s, differ. The Hedden-Ording examples have nontrivial Alexander polynomials and are not topologically slice.…

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(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…

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots…

In this paper we examine the relationship between various types of positivity for knots and the concodance invariant tau discovered by Ozsvath and Szabo and independently by Rasmussen. The main result shows that, for fibered knots, tau…

In this paper, we introduce a rational $\tau$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsv\'{a}th-Szab\'{o} contact invariants. Such an invariant is an upper bound for the sum of rational…

Using a knot concordance invariant from the Heegaard Floer theory of Ozsvath and Szabo, we obtain new bounds for the Thurston-Bennequin and rotation numbers of Legendrian knots in S^3. We also apply these bounds to calculate the knot…

Ozsvath and Szabo defined an analog of the Froyshov 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…

We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.

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…

Let $\nu$ be either the Ozsv\'ath-Szab\'o $\tau$-invariant or the Rasmussen $s$-invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_\nu(K)$ to be the minimum of $k$ for which $\nu$ of the $k$-twisted…

We derive new existence results for tight contact structures on certain 3-manifolds which can be presented as surgery along specific knots in S^3. Indeed, we extend our earlier results on knots with maximal Thurston-Bennequin number being…

The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now…

We prove a cabling formula for the concordance invariant $\nu^+$, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.

For any rational homology 3-sphere and one of its spin^{c}-structures, Ozsvath and Szabo defined a topological invariant, called d-invariant. Given a knot in the 3-sphere, the d-invariants associated with the prime-power-fold branched…

The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining…

In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsvath and Szabo. Along the way, we prove that if two knots are concordant, then their knot Floer complexes…

Ozsv\'ath and Szab\'o used the knot filtration on $\widehat{CF}(S^3)$ to define the $\tau$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $\tau$-invariants associated to a…