Strongly Quasipositive Links

• Strongly quasipositive links are defined as oriented links that arise as closures of braids built from positive embedded bands, equivalent to boundaries of quasipositive Seifert surfaces.
• They exhibit robust invariance properties such as closure under Murasugi sums and strict genus formulas that link classical invariants like self-linking and the Conway polynomial.
• Their application spans braid theory, contact topology, and complex geometry, providing key insights into fibered links, transverse knots, and the classification of almost positive links.

Strongly quasipositive links are oriented links that arise as closures of braids built from positive embedded bands, equivalently as boundaries of quasipositive Seifert surfaces obtained from parallel disks by attaching only positively twisted bands. In braid-theoretic, surface-theoretic, and contact-geometric terms, strong quasipositivity is a refinement of positivity, while in geometric terms it captures links bounded by a particularly rigid class of surfaces. This class appears naturally in the study of complex plane curves, transverse links, open books, and fibered links, and it interacts sharply with classical invariants such as genus, self-linking number, the Conway polynomial, and Heegaard Floer and Khovanov-type concordance invariants (Abe et al., 2014, Silvero, 2015, Hayden, 2017).

1. Foundational definitions and surface models

Let $B_n$ be the braid group with Artin generators $\sigma_1,\dots,\sigma_{n-1}$. One standard family of positive embedded bands is

$$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$

In Birman–Ko–Lee notation, the same objects are often written $a_{i,j}$. A braid is strongly quasipositive if it is a product of such positive embedded bands, and a link is strongly quasipositive if it is the closure of a strongly quasipositive braid (Abe et al., 2014, Truöl, 2022).

The surface model is equivalent and is often more useful. If a braid on $n$ strands is written as a product of $m$ positive embedded bands, the associated quasipositive surface is obtained from $n$ parallel disks by attaching $m$ positively twisted bands, one for each factor. Its Euler characteristic is $n-m$. Rudolph’s interpretation identifies strongly quasipositive links exactly with boundaries of such quasipositive Seifert surfaces (Abe et al., 2014, Silvero, 2015).

This geometric formulation is stable under several natural operations. Murasugi sums of quasipositive Seifert surfaces are quasipositive, and incompressible subsurfaces of quasipositive Seifert surfaces are quasipositive (Feller et al., 2018). These closure properties are fundamental in inductive constructions and in proofs that large diagrammatic classes produce strongly quasipositive links.

2. Relation to positivity and diagrammatic criteria

Strong quasipositivity sits in a standard hierarchy of positivity notions: $$\text{positive braid link} \;\Rightarrow\; \text{positive link} \;\Rightarrow\; \text{strongly quasipositive link} \;\Rightarrow\; \text{quasipositive link}.$$ The first two implications are classical, and positive links are strongly quasipositive by work of Rudolph and Nakamura (Abe et al., 2014, Hamer et al., 2018).

The class is strictly larger than the class of positive links. A concrete family is

$\sigma_1,\dots,\sigma_{n-1}$0

These links are strongly quasipositive by construction, and for even $\sigma_1,\dots,\sigma_{n-1}$1 they satisfy $\sigma_1,\dots,\sigma_{n-1}$2; the paper shows that in this case they are not positive (Silvero, 2015). This distinction is central: strong quasipositivity is not merely positivity in disguise.

A major extension of the positive world is that every almost positive link is strongly quasipositive. Here “almost positive” means that the link admits a diagram with exactly one negative crossing. The proof splits almost positive diagrams into two types, according to whether the unique negative crossing has a parallel positive crossing, and in both cases constructs a quasipositive surface bounded by the link (Feller et al., 2018). This settles Stoimenow’s question in the strong positive.

For canonical Seifert surfaces, there is a precise graph-theoretic test. If $\sigma_1,\dots,\sigma_{n-1}$3 is an oriented diagram and $\sigma_1,\dots,\sigma_{n-1}$4 its Seifert graph, then the canonical surface $\sigma_1,\dots,\sigma_{n-1}$5 is quasipositive if and only if every cycle in $\sigma_1,\dots,\sigma_{n-1}$6 has strictly positive total weight (Feller et al., 2018). In particular, for canonical surfaces, quasipositivity is encoded by weighted Seifert-graph combinatorics rather than by an a priori braid description.

Among homogeneous links, strong quasipositivity becomes even more rigid: positive links are exactly the homogeneous strongly quasipositive links (Abe et al., 2014). In the alternating setting, Orevkov proved that if an alternating diagram is a Diao–Hetyei–Liu diagram, equivalently no pair of Seifert circles is connected by a single crossing, then quasipositivity forces the diagram to be positive, hence the link is strongly quasipositive (Orevkov, 2020).

3. Fiberedness, open books, and contact geometry

Strong quasipositivity is particularly effective in fibered and open-book settings. In the Birman–Ko–Lee framework, if a strongly quasipositive braid has dual Garside normal form

$\sigma_1,\dots,\sigma_{n-1}$7

with $\sigma_1,\dots,\sigma_{n-1}$8, then its closure is fibered (Banfield, 2016). The same paper identifies this class geometrically as the boundaries of plumbings of positive Hopf bands to a disk, equivalently $\sigma_1,\dots,\sigma_{n-1}$9 Hopf-plumbed baskets. This gives a large and explicit fibered subclass of strongly quasipositive links (Banfield, 2016).

A contact-geometric characterization replaces braid positivity by a ribbon condition. In an arbitrary open book, a link is strongly quasipositive if and only if it bounds a Legendrian ribbon in the associated contact structure (Hayden, 2017). This generalizes the $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$0 result of Baader and Ishikawa and shows that strong quasipositivity is intrinsic to the contact-open-book package, not merely to a particular braid word.

For non-fibered links, the fibered equivalence between strong quasipositivity and tightness breaks. Nevertheless, strongly quasipositive links still induce tight contact structures via the partial open books associated to incompressible quasipositive Seifert surfaces. The converse fails: there are non-fibered links that support tight contact structures but are not strongly quasipositive, with the Stevedore knot $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$1 serving as the basic example (Nonino et al., 30 Sep 2025).

The fibered case remains exceptionally rigid. For fibered links in $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$2, the background cited in the non-fibered study states that being strongly quasipositive and supporting a tight contact structure are equivalent notions (Nonino et al., 30 Sep 2025). This rigidity is one reason fibered strongly quasipositive links occupy a distinguished position in contact topology.

4. Genus, self-linking, and polynomial invariants

Non-split strongly quasipositive links satisfy a strong genus formula. If $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$3 is non-split and strongly quasipositive, and $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$4 is a quasipositive surface for $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$5, then

$$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$6

where $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$7 is the Rasmussen–Beliakova–Wehrli invariant, $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$8 the smooth $$\sigma_{i,j}=(\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})\, \sigma_{j-1}\, (\sigma_i \sigma_{i+1} \cdots \sigma_{j-2})^{-1}, \qquad 1\le i<j\le n.$$9-ball genus, and $a_{i,j}$0 the Seifert genus (Abe et al., 2014). Thus strong quasipositivity forces equality of the three-genus and four-ball genus.

For knots whose canonical genus equals the genus, strong quasipositivity is equivalent to sharpness of the Bennequin inequality and also equivalent to equality $a_{i,j}$1 for a slice–torus invariant $a_{i,j}$2. In this regime, canonical-surface quasipositivity, strong quasipositivity, maximal self-linking, and slice–torus saturation coincide (Feller et al., 2018).

The defect of the Bennequin inequality gives another viewpoint. Strongly quasipositive links satisfy $a_{i,j}$3, and for braid index $a_{i,j}$4 this condition is exact: if $a_{i,j}$5, then sharpness of the Bennequin inequality, strong quasipositivity, and existence of a strongly quasipositive $a_{i,j}$6-braid representative are equivalent (Hamer et al., 2018). In homogeneous families, the same sharpness collapses to positivity and hence to strong quasipositivity (Hamer et al., 2018, Abe et al., 2014).

Classical polynomial invariants also detect structure. If $a_{i,j}$7 is a strongly quasipositive link with braid index $a_{i,j}$8, then its Conway polynomial has non-negative coefficients. This gives a necessary condition for strong quasipositivity in braid index $a_{i,j}$9, and the paper proves that the phenomenon is special to that braid index by exhibiting a strongly quasipositive $n$0-braid whose closure has Conway polynomial $n$1 (Silvero, 2015).

Branched-cover constraints are stronger still. If $n$2 is strongly quasipositive and some cyclic branched cover $n$3 is an L-space, then $n$4 is definite, $n$5, all roots of $n$6 lie in a prescribed arc $n$7, and in the monic knot case one has $n$8 (Boileau et al., 2017). For strongly quasipositive knots with monic Alexander polynomial, no cyclic branched cover of degree $n$9 can be an L-space (Boileau et al., 2017).

5. Specialized families and classification results

Several structured subclasses of strongly quasipositive links admit precise classification.

Checkerboard graph links form a class of fibred strongly quasipositive links that includes positive braid links. For this family, maximal signature forces ADE rigidity: a checkerboard graph link with maximal signature is isotopic to one of the links realized by the simply laced Dynkin diagrams $m$0, $m$1, $m$2, $m$3, or $m$4 (Vilanova, 2019).

In the quasi-alternating setting, definiteness can characterize strong quasipositivity. If $m$5 is quasi-alternating with a quasi-alternating crossing $m$6 such that $m$7 is alternating, then $m$8 is definite if and only if it is strongly quasipositive, up to mirroring. The same equivalence holds when $m$9 is fibred, or more generally has a unique minimal genus Seifert surface (Ba, 2019).

Montesinos and pretzel families exhibit both positive and negative results. The quasi-alternating study produces explicit strongly quasipositive Montesinos families and explicit non-strongly-quasipositive Montesinos families via continued-fraction conditions (Ba, 2019). The branched-cover paper gives a complete classification of strongly quasipositive $n$0-strand pretzel knots $n$1 under the stated parity and sign conditions, and corresponding results for oriented $n$2-strand pretzel links (Boileau et al., 2017).

A recent structured subclass is furnished by $n$3-positive links. These form a subset of strongly quasipositive links strictly containing all non-split braid positive links, and they are precisely the strongly quasipositive links that are closures of $n$4-homogeneous braids. They coincide with boundaries of positive Hopf-plumbed baskets and with closures of staircase braids (Bode et al., 11 May 2026). In particular, all $n$5 strongly quasipositive, fibered knots with at most $n$6 crossings are $n$7-positive (Bode et al., 11 May 2026).

For $n$8-space links, the Floer-theoretic notion of a special $n$9-space link singles out a fibered strongly quasipositive class. Every special $m$0-space link is fibered and strongly quasipositive, and for $m$1-component $m$2-space links this condition is also necessary (Cavallo et al., 2020).

6. Concordance and current directions

Strong quasipositivity is not rigid under smooth concordance in the non-fibered category. Every strongly quasipositive link other than an unlink is smoothly concordant to infinitely many pairwise non-isotopic strongly quasipositive links. The construction uses a satellite operation with companion a slice knot of maximal Thurston–Bennequin number $m$3, and it contrasts sharply with the conjectural rigidity picture for fibered strongly quasipositive knots (Truöl, 2022).

Several open problems organize current research. One direction asks for stronger braid-polynomial criteria. A combinatorial characterization of equality in the Morton–Franks–Williams bound is known for positive diagrams, and the same work proposes an analogue for a large class of strongly quasipositive braids built from the trivial braid by adding squares of positive band generators, doublings of band generators, and Birman–Ko–Lee moves (Alekseev, 2022). Another direction asks whether almost positive links can be characterized as strongly quasipositive links satisfying an “almost homogeneous” condition, mirroring the characterizations of positive and $m$4-positive links (Bode et al., 11 May 2026).

In the alternating and contact-topological settings, the main unresolved questions are equally structural. Orevkov asks whether the inequality $m$5 holds for every reduced alternating diagram; an affirmative answer would force broad classes of alternating quasipositive links to be positive and hence strongly quasipositive (Orevkov, 2020). In the non-fibered contact setting, the partial-open-book theory asks whether every tight partial open book arising from an incompressible Seifert surface is modeled by a strongly quasipositive surface, and whether associated contact invariants must be nonzero for strongly quasipositive links (Nonino et al., 30 Sep 2025).

The modern picture is therefore two-sided. On one side, strong quasipositivity admits exact braid, surface, graph, Floer-theoretic, and contact-geometric characterizations in several important regimes. On the other, recent work shows that the class is broad enough to support rich concordance phenomena, subtle failures of polynomial positivity beyond braid index $m$6, and nontrivial distinctions from positivity, homogeneity, fiberedness, and tightness.