Notions of Positivity and the Ozsváth–szabó Concordance Invariant

Abstract

In this paper we examine the relationship between various types of positivity for knots and the concodance invariant τ discovered by Ozsváth and Szabó and independently by Rasmussen. The main result shows that, for fibered knots, τ characterizes strong quasipositivity. This is quantified by the statement that for K fibered, τ (K) = g(K) if and only if K is strongly quasipositive. In addition, we survey existing results regarding τ and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive. For instance, we show that any knot known to admit a lens space surgery is strongly quasipositive and exhibit infinite families of knots which are not quasipositive.cc

The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal

“…See also [32]. (4) Quasipositive knots: s(K) = 2τ (K) = 2g 4 (K), where g 4 (K) denotes the smooth slice genus of K. This follows from work of Plamenevskaya [27] for τ and from Plamenevskaya [28] and Shumakovitch [34] for s. See also [7]. (5) Knots with up to 10 crossings.…”

Fibered Transverse Knots and the Bennequin Bound

“…So this corollary applies to fibered positive knots. While the appearance of strongly quasi-positive links may appear mysterious in this corollary, it is really quite natural given Corollary 1.2 above and Hedden's observation [16] that that a fibered link in S 3 supports the standard tight contact structure on S 3 if and only if it is strongly quasi-positive. We now explore some consequence of our theorems above to problems in braid theory.…”

“…Since that time there has been much work finding other bounds for transverse knots in S 3 [9,14,12] and some work finding other bounds in general manifolds [16,22]. For some links, other bounds are more restrictive and the Bennequin inequality is not sharp.…”

“…In general there is little geometric understanding of the knot types for which the Bennequin inequality is sharp. Building on an interesting result of Hedden [16] discussed below, our first goal in this paper is to rectify this situation for fibered links by giving a specific necessary and sufficient condition for sharpness. Once we have done this we will discuss several corollaries concerning the classification of transversal knots, braids representing quasi-positive links and the determination of a contact structure via its transverse knot theory.…”

Lens space surgeries as A’Campo’s divide knots

“…[33] that every knot admitting a lens space surgery is fibered rather than just that Berge knots are fibered; see also Hedden [25]. Their argument does not rely upon double primitivity.…”