On overtwisted, right-veering open books

Abstract

We exhibit infinitely many overtwisted, right-veering, non-destabilizable open books, thus providing infinitely many counterexamples to a conjecture of Honda, Kazez and Matić. The page of all our open books is a four-holed sphere and the underlying 3-manifolds are lens spaces.

Visualizing Overtwisted Discs in Open Books

“…This negatively answers a question of Honda, Kazez and Matić [14]. Our family generalizes the previously known examples by Lekili [19] and Lisca [20], but our proof of overtwistedness is more direct.…”

“…In [14, Question 6.2] Honda, Kazez and Matić ask whether a right-veering and nondestabilizable open book always supports a tight contact structure. Lekili [19] and Lisca [20] negatively answer the question by constructing examples. They study open book decompositions of 3-manifolds whose tight contact structures are well-studied and classified (in [19] Poincaré homology 3-spheres, and in [20] lens spaces).…”

“…Lekili [19] and Lisca [20] negatively answer the question by constructing examples. They study open book decompositions of 3-manifolds whose tight contact structures are well-studied and classified (in [19] Poincaré homology 3-spheres, and in [20] lens spaces). In both constructions the most technical points are showing that their open books indeed support overtwisted 6 Proof.…”

Fractional Dehn twists in knot theory and contact topology

“…Lisca [18] constructed an infinite number of counterexamples for the same surface. Ito and Kawamuro [16] have produced an even larger set of counterexamples on the 4 times punctured sphere.…”

“…If true, this conjecture would provide the simplest known counterexamples to a conjecture of Honda, Kazez and Matić [14] by producing fibred knots in S 3 that are not stabilizations, are right-veering, yet are still overtwisted. The section includes some context for the conjecture of [14], references to earlier counterexamples for links bounding planar surfaces by Lekili [17], Lisca [18] and Ito and Kawamuro [16], and some remarks on work of Colin and Honda [5]. The technique we use for recognizing that a contact structure is overtwisted is completely elementary: we exhibit an overtwisted disk by finding an unknotted, untwisted curve on the Seifert surface of the knot.…”