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Non-fibered strongly quasipositive links and tightness

Isacco Nonino, Miguel Orbegozo Rodriguez

TL;DR

This work investigates the relationship between strong quasipositivity and tightness for non-fibered links by harnessing Gabai's product disks to form partial open books and the induced contact structures on manifolds with convex boundary. It proves that if a link is strongly quasipositive, the associated partial open book is right-veering and supports a tight contact structure, extending Hedden's fibered-case result beyond fibered links. Conversely, tightness of the supported partial open book does not imply strong quasipositivity, as shown by the knot $6_1$ and an infinite family of examples; these constructions reveal subtle distinctions between fibered and non-fibered cases. The surfaces arising in these examples are abstractly diffeomorphic to Seifert surfaces of strongly quasipositive knots with the same product disks, and the paper ends with questions about when an abstract partial open book corresponds to a Seifert surface of a strongly quasipositive link and the implications for Giroux torsion.

Abstract

It is well known that for fibered links in $\mathbb{S}^3$ being strongly quasipositive and supporting a tight contact structure are equivalent notions (arXiv:math/0509499). In this note we analyze the relation between these two properties for non fibered links. A non fibered link (together with an incompressible Seifert surface) induces a natural partial open book (arXiv:2509.09615). We prove that strongly quasipositive links induce tight contact structures. We also show that, in contrast to the fibered case, the converse is not true, giving examples of links that are not strongly quasipositive but support tight contact structures.

Non-fibered strongly quasipositive links and tightness

TL;DR

This work investigates the relationship between strong quasipositivity and tightness for non-fibered links by harnessing Gabai's product disks to form partial open books and the induced contact structures on manifolds with convex boundary. It proves that if a link is strongly quasipositive, the associated partial open book is right-veering and supports a tight contact structure, extending Hedden's fibered-case result beyond fibered links. Conversely, tightness of the supported partial open book does not imply strong quasipositivity, as shown by the knot and an infinite family of examples; these constructions reveal subtle distinctions between fibered and non-fibered cases. The surfaces arising in these examples are abstractly diffeomorphic to Seifert surfaces of strongly quasipositive knots with the same product disks, and the paper ends with questions about when an abstract partial open book corresponds to a Seifert surface of a strongly quasipositive link and the implications for Giroux torsion.

Abstract

It is well known that for fibered links in being strongly quasipositive and supporting a tight contact structure are equivalent notions (arXiv:math/0509499). In this note we analyze the relation between these two properties for non fibered links. A non fibered link (together with an incompressible Seifert surface) induces a natural partial open book (arXiv:2509.09615). We prove that strongly quasipositive links induce tight contact structures. We also show that, in contrast to the fibered case, the converse is not true, giving examples of links that are not strongly quasipositive but support tight contact structures.

Paper Structure

This paper contains 7 sections, 4 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Acknowledgments
  4. Preliminaries
  5. Fiberedness and strong quasipositivity
  6. Partial open books and contact manifolds with convex boundary
  7. Constructions and results

Key Result

Lemma 3.2

Let $K$ be a link, $S$ an incompressible Seifert surface for $K$. Then $(S,P,h)$ is a well defined partial open-book associated to the pair $(S,K)$.

Figures (2)

  • Figure 1: A Murasugi sum of two surfaces $\Sigma_1$ and $\Sigma_2$ along $P$. Figure taken from Murasugisums.
  • Figure 2: The Pretzel Knot $(-3,3,1)$, which is the Stevedore knot $6_1$.

Theorems & Definitions (23)

  • Definition 2.1: Fibered link
  • Remark 2.2
  • Definition 2.3: Nearly fibered knot and almost fibered knot
  • Definition 2.3: Nearly fibered knot and almost fibered knot
  • Remark 2.4
  • Definition 2.5: Quasipositive surface and strongly quasipositive links
  • Definition 2.6: Murasugi sum
  • Definition 2.7: Partial open book decompositions
  • Remark 2.8
  • Remark 2.8
  • ...and 13 more