arXiv subjects

John B. Etnyre

Publications and source records attributed to John B. Etnyre.

At least 19 recordsLinked to original sources

arXiv2026

Tight surgeries on torus knots

We give a complete classification of tight contact structures on manifolds obtained from any surgery along any torus knot, which provides the first such classification on an infinite family of knots. The main tools are the classification of non-loose torus knots, bordered contact invariants via their immersed curve interpretations, and non-thickenable tori in knot complements. As a byproduct of this classification, we also give the first classification of contact structures on some surface bundles over the circle with fibers of genus larger than $1$. The symplectic fillability of these contact structures is also studied and leads to, among other things, the disproof of a conjecture of Ghiggini and Van Horn-Morris on Stein fillability of some contact structures on a Brieskorn sphere.

math.GT↗

arXiv2026

Complementary legs and symplectic rational balls

We show that a small Seifert fibered space with complementary legs does not symplectically bound a rational homology ball for at least one choice of orientation. In the case $e_0\leq -1$, we characterize when a small Seifert fibered space with uniquely complementary legs symplectically bounds a rational homology ball. In the case $e_0\geq 0$, we characterize when a small Seifert fibered space with complementary legs, equipped with a balanced contact structure, symplectically bounds a rational homology ball. Our results highlight a sharp contrast with the smooth category, where many more such Seifert fibered spaces are known to bound smooth rational homology balls. As a consequence of the results above, we also complete the classification of contact structures on oriented spherical $3$-manifolds that admit symplectic rational homology ball fillings. In particular, we show that a closed, oriented $3$-manifold with finite fundamental group admits at most six contact structures, up to isotopy, which are symplectically fillable by rational homology balls.

math.GT↗

arXiv2026

Non-loose torus knots

We give a complete coarse classification of Legendrian and transverse torus knots in any contact structure on $S^3$.

math.GT↗

arXiv2026

On contact cosmetic surgery

We demonstrate that the contact cosmetic surgery conjecture holds true for all non-trivial Legendrian knots, with the possible exception of Lagrangian slice knots. We also discuss the contact cosmetic surgeries on Legendrian unknots and make the surprising observation that there are some Legendrian unknots that have a contact surgery with no cosmetic pair, while all other contact surgeries are contactomorphic to infinitely many other contact surgeries on the knot.

math.GT↗

arXiv2026

Neighborhoods of transverse knots and destabilizations

In this note, we show that transverse knots have unique standard neighborhoods and prove a structure theorem about non-loose Legendrian knots. We also prove a finiteness result for transverse knots in a tight contact manifold. The common theme of these two results is a general destabilization result for Legendrian knots. As a byproduct of this work, we find a manifold with an infinite number of distinct tight contact structures, up to contactomoprhism, with no Giroux torsion.

math.GT↗

arXiv2026

Building homology theories (ala Floer)

These notes are an expanded version of evening talks at the 2025 Georgia International Topology Conference, and an abbreviated version of talks at Georgia Tech, which were aimed at graduate students. The hope was to indicate a common framework that has been used since the late 1980s to construct homology theories in low-dimensional topology and symplectic and contact geometry. In addition to this, we also try to indicate how the specific nature of the situation being studied dictates the algebraic nature of the chain groups used to define the homology.

math.GT↗

arXiv2025

Submanifolds in contact geometry

We survey what is known about various special types of submanifolds of contact manifolds and discuss their role in the development of contact geometry.

math.SG↗

arXiv2025

Symplectic rational homology ball fillings of Seifert fibered spaces

We characterize when some small Seifert fibered spaces can be the convex boundaries of symplectic rational homology balls and give strong restrictions for others to bound such manifolds. In particular, we show that the only spherical $3$-manifolds, oriented as links of the corresponding quotient singularities, which admit symplectic rational homology ball fillings, are the lens spaces $L(p^2,pq-1)$ previously identified by Lisca. In a different direction, we provide evidence for Gompf's conjecture that Brieskorn spheres do not bound Stein domains in $\mathbb{C}^2$. Finally, we establish restrictions on Lagrangian disk fillings of certain Legendrian knots in small Seifert fibered spaces.

math.GT↗

arXiv2025

Small symplectic caps and embeddings of homology balls in the complex projective plane

We present a handlebody construction of small symplectic caps, and hence of small closed symplectic 4-manifolds. We use this to construct handlebody descriptions of symplectic embeddings of rational homology balls in $\mathbb{C}\mathrm{P}^2$, and thereby provide the first examples of (infinitely many) symplectic handlebody decompositions of a closed symplectic 4-manifold. Our constructions provide a new topological interpretation of almost toric fibrations of $\mathbb{C}\mathrm{P}^2$ in terms of symplectic handlebody decompositions.

math.GT↗

arXiv2025

Cable links of uniformly thick knot types

In this paper, we study Legendrian realizations of cable links of knot types that are uniformly thick but not Legendrian simple, extending prior work of Dalton, the second author, and Traynor. This leads to new phenomena, such as stabilized Legendrian links that are smoothly isotopic and component-wise Legendrian isotopic, but are not Legendrian isotopic. In our study, we establish new results for cable links whose cabling slope is sufficiently negative. We will also show how to classify Legendrian knots in (most) negative cables of twist knots. This is done by introducing a new technique to the study of cables based on Legendrian surgeries.

math.GT↗

arXiv2025

Tight and overtwisted contact structures

The tight versus overtwisted dichotomy has been an essential organizing principle and driving force in 3-dimensional contact geometry since its inception around 1990. In this article, we will discuss the genesis of this dichotomy in Eliashberg's seminal work and his influential contributions to the theory.

math.SG↗

arXiv2024

Lagrangian Realizations of Ribbon Cobordisms

We show that up to stabilizations, smooth ribbon cobordisms can be realized by decomposable Lagrangian cobordisms. We also define the notion of stabilization for Lagrangian cobordisms and show that it can be used to find new Lagrangian cobordisms between Legendrian links.

math.SG↗

arXiv2023

Existence and construction of non-loose knots

In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and discuss conditions under which non-looseness is preserved under cabling.

math.GT↗

arXiv2023

Legendrian Torus and Cable Links

We give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give the first family of links that are non-destabilizable but do not have maximal Thurston-Bennequin invariant and observe a curious distribution of Legendrian torus knots that can be realized as the components of a Legendrian torus link. This classification of Legendrian torus links leads to a classification of transversal torus links. We also give a classification of Legendrian and transversal cable links of knot types that are uniformly thick and Legendrian simple. Here we see some similarities with the classification of Legendrian torus links but also some differences. In particular, we show that there are Legendrian representatives of cable links of any uniformly thick knot type for which no symmetries of the components can be realized by a Legendrian isotopy, others where only cyclic permutations of the components can be realized, and yet others where all smooth symmetries are realizable.

math.GT↗

arXiv2023

Legendrian contact homology in $\mathbb{R}^3$

This is an introduction to Legendrian contact homology and the Chekanov-Eliashberg differential graded algebra, with a focus on the setting of Legendrian knots in $\mathbb{R}^3$. This is the published version of the paper, but with a section of errata added at the end.

math.SG↗

arXiv2022

Generalizations of planar contact manifolds to higher dimensions

Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer m > 3, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension m. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.

math.SG↗

arXiv2022

Symplectic hats

We study relative symplectic cobordisms between contact submanifolds, and in particular relative symplectic cobordisms to the empty set, that we call hats. While we make some observations in higher dimensions, we focus on the case of transverse knots in the standard 3-sphere, and hats in blow-ups of the (punctured) complex projective planes. We apply the construction to give constraints on the algebraic topology of fillings of double covers of the 3-sphere branched over certain transverse quasipositive knots.

math.GT↗

arXiv2022

Transverse universal links

We show that there exists a transverse link in the standard contact structures on the 3-sphere such that all contact 3-manifolds are contact branched covers over this transverse link.

math.GT↗