Slice-torus Concordance Invariants and Whitehead Doubles of Links

Abstract

In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant.2010 Mathematics Subject Classification. 57M27. 1 These function were originally defined only for integer-valued slice-torus invariants. Of course, the same definition works for all slice-torus invariants, and most of the properties proved in [23] still hold. 1 2 ALBERTO CAVALLO AND CARLO COLLARIwhere ν is a slice-torus invariant and W ± (K, t) denotes the positive (resp. negative) ttwisted Whitehead double of K. These functions are non-increasing, non-constant, take values respectively in [0, 1] and [−1, 0], and assume both the maximal and the minimal possible values. In particular, if the slice-torus invariant is integer-valued all the information contained in each function can be condensed into a single integer. These integers, denoted by t ν and t ν , are defined as the maximal value of t such that F ν (K, t) and F ν (K, t), respectively, assume their maximum. It is not difficult to see that t ν (K) = −t ν (−K * ) − 1, where −K * is the mirror image of K with the orientation reversed, so these invariants contain the same amount of information. At the time of writing it is still unknown whether the invariant t ν can provide new information with respect to ν. In fact there are some hints in the opposite direction; for instance, it is known that t τ = 2τ − 1 ([15, Theorem 1.5]) and it has been conjectured that t s/2 = 3s/2 − 1 ([29]).The aim of the present paper is to extend these definitions and constructions to the case of links, and to describe some applications and examples. Before stating the main results of this paper let us recall a few basic facts about link concordance. The first thing one should point out is that the definition of concordance is no longer unique. Two oriented links in S 3 are said to be ⊲ weakly concordant if there exists a genus 0 connected, compact, oriented surface, properly embedded in S 3 × [0, 1], bounding the two links; ⊲ strongly concordant if there exists a disjoint union of annuli, properly embedded in S 3 × [0, 1], such that each of them bounds a component of each link. In particular, strongly concordant links should have the same number of components. A link is said to be weakly (resp. strongly) slice if is weakly (resp. strongly) concordant to an unlink. Similarly, one can define a (weak ) slice genus and a strong slice genus. The former is just the minimal genus of any connected, compact, oriented surface properly embedded in S 3 × [0, 1] bounding the link. The latter has a similar definition but one has t...

Slice‐torus link invariants, combinatorial invariants and positivity conditions

“…[11, Proposition 2.7]), therefore . Since , by [10, Proposition 2.11], we obtain the desired equality. Finally, has two trivial components with linking number zero.…”

“…The situation is different if one looks from the smooth perspective. Firstly, we recall that slice‐torus link invariants are a special class of ‐valued concordance invariants of links, which vanish on unlinks — see [10] for the precise definition. Prominent examples of slice‐torus link invariants are the Ozsváth–Szabó–Rasmussen ‐invariant [7, 27, 29] and (a re‐scaling of) the Rasmussen ‐invariant [3, 30].…”

“…Prominent examples of slice‐torus link invariants are the Ozsváth–Szabó–Rasmussen ‐invariant [7, 27, 29] and (a re‐scaling of) the Rasmussen ‐invariant [3, 30]. It is known that the slice‐torus link invariants provide an obstruction to being concordant to a quasi‐positive link; indeed, it follows from [10, Theorem 3.2] (see [9, 21], and references therein, for knots, and special slice‐torus link invariants) that: if is an ‐component link concordant to a quasi‐positive link, and is a slice‐torus invariant, then where is the (smooth) slice Euler characteristic , that is, An immediate consequence of (1.1) is that there are links which are not concordant to any quasi‐positive link, for example, the figure‐eight knot is such that , and [22].…”

“…Firstly, we give an example where (1.1) provides a stronger obstruction than Theorem 1.3. Recall that is the split union of , we write , if and the links are separated by embedded copies of .Example Slice‐torus link invariants are additive under split union [10, Definition 2.1 (B) ]. Thus, the split union of a figure‐eight knot and an unlink does not satisfy (1.1), but satisfies (1.2).…”

“…Proof Using the diagram in Figure 1, and [10, Theorem 1.3] (cf. Proposition 2.4), one can compute .…”