The paper defines and studies contact surgery numbers for contact 3-manifolds.
problem Understanding the minimal number of components of a surgery link describing a contact 3-manifold.
method Defined and studied various versions of contact surgery numbers, relating them to other invariants and computing specific cases.
result There exist infinitely many non-isotopic contact structures on certain manifolds that cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere.
Study confirms contact cosmetic surgery for most knots, with exceptions.
problem Determining contact cosmetic surgeries for Legendrian knots.
method Analyzing Legendrian knots, including unknots, and using contact surgery theory.
result Some Legendrian unknots have unique contact surgeries without a cosmetic pair.
New result on contact surgeries and overtwistedness.
problem When contact (+1/n)-surgery is overtwisted.
method Analyzes contact surgeries and provides counterexamples.
result Provides conditions for overtwisted contact surgeries.
Contact round surgeries on (S3,ξst) help in constructing and understanding contact 3-manifolds.
problem Constructing contact 3-manifolds using Legendrian surgeries.
method Introducing contact round surgeries of indices 1 and 2, and associating them with surgery diagrams.
result Every closed connected contact 3-manifold can be obtained by a sequence of contact round surgeries on Legendrian knots in (S3,ξst). Defines contact surgery distance and shows it's bounded by topological surgery distance by 5.
problem Comparing contact structures on 3-manifolds.
method Defining contact surgery distance and proving upper bound on its value.
result Contact surgery distance is at most 5 larger than topological surgery distance.
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
Graphs describe contact surgery on 3-manifolds.
problem Understanding contact surgery on 3-manifolds.
method Defined contact surgery graphs to analyze their properties.
result Analyzed basic properties and interesting subgraphs of contact surgery graphs.
Study contact structures on 3-torus via surgery, proving non-contactomorphic cases and overtwisted structures.
problem Characterizing contact structures on 3-torus via surgery.
method Contact surgery, Legendrian surgery, ruling invariant, Thurston-Bennequin invariant.
result Proves infinitely many pairwise non-contactomorphic contact structures on 3-torus.
Study contact surgeries on Legendrian links, proving invariant vanishing and overtwistedness.
problem Understanding contact surgeries on Legendrian links and their properties.
method Algebraic methods using Heegaard Floer homology and contact-geometric arguments.
result Vanishing of contact Ozsváth-Szabó invariant and overtwistedness of contact 3-manifolds.
Contact surgeries transform manifolds, and all admit symplectic caps.
problem Transforming contact manifolds using surgeries.
method Contact surgeries on isotropic and coisotropic spheres.
result All closed oriented contact manifolds admit symplectic caps.
Note on potential Kirby move type 1 for contact surgery diagrams.
problem Exploring conditions for a contact Kirby move type 1.
method Analyzing necessary conditions for contact surgery diagrams to be candidates for contact Kirby move type 1.
result Existence of a collection of contact positive integral surgery diagrams on Legendrian unknots satisfying the conditions.
Surgery on knots always admits a tight contact structure.
problem Understanding tight contact structures on knots after surgery.
method Smooth (-r)-surgery on knots, using Heegaard Floer contact invariant.
result Tight contact structures detected for all knots after surgery.
Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.
problem Classifying contact structures with specific contact surgery numbers on Brieskorn spheres and lens spaces.
method Algorithm for computing Euler class from rational contact surgery, Legendrian knot classification, and analysis of contact structures.
result Infinitely many non-isotopic contact structures on Brieskorn spheres and lens spaces cannot be obtained by a single rational contact surgery.
Contact surgeries yield algebraically overtwisted manifolds.
problem Understanding algebraically overtwisted contact manifolds through surgeries.
method Contact (+1)-surgeries on Legendrian spheres in flexibly fillable contact manifolds. result Yielding algebraically overtwisted manifolds when the Legendrian's homology class is not annihilated.
In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact plus or minus 1 surgeries, and used this to prove that any (closed) contact…
Generalizes surgery techniques for projectively Anosov flows.
problem Creating new projectively Anosov flows from existing ones.
method Introducing a generalized Goodman surgery technique for projectively Anosov flows.
result Generates new examples of projectively Anosov flows on hyperbolic 3-manifolds.
Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard con…
For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery. In ad…
We construct an open book decomposition compatible with a contact structure given by a rational contact surgery on a Legendrian link in the standard contact S3. As an application we show that some rational contact surgeries on certain Legendrian knots induce overtwisted contact structures.
New surgeries on knots preserve contact structures.
problem Understanding how surgeries on Legendrian knots affect their contact structures.
method Analyzing surgeries on specific types of knots (twist and two-bridge knots) and proving distinct contact structures for certain surgeries.
result Negative rational surgeries on certain Legendrian knots yield distinct contact 3-manifolds.
We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. The overarching theme of this paper is to show that in many contexts, surgery on transverse knots is more natural than s…
The paper solves when surgeries on Legendrian knots yield symplectically fillable contact manifolds.
problem When contact surgeries on Legendrian knots yield symplectically fillable contact manifolds.
method Analyzes contact (r)-surgeries on Legendrian knots and investigates Lagrangian fillings.
result Completely answers the question of symplectically fillable contact manifolds after contact surgeries.
The study finds algebraically overtwisted tight 3-manifolds via contact surgeries.
problem Finding algebraically overtwisted tight 3-manifolds.
method Executing Avdek's algorithm to perform contact surgeries.
result Contact surgeries on standard contact 3-sphere yield algebraically overtwisted and tight 3-manifolds.
Let (M,ξ) be a contact 3-manifold. We present two new algorithms, the first of which converts an open book (Σ,Φ) supporting (M,ξ) with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for (M,ξ) into a supporting open book decomposition. These constructions lead to a r…
We give new tightness criteria for positive surgeries along knots in the 3-sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matic's, Ozsvath and Szabo's Floer-theoretic contact invariants. We compute the Ozsvath and Szabo's invariant of positive contact surgeries …
New surgery method preserves Anosov flow properties using bi-contact geometry.
problem Defining a new type of surgery on Anosov flows.
method Bi-contact geometry and Reeb dynamics.
result Necessary and sufficient condition for generating contact Anosov flows.
We study cosmetic contact surgeries along transverse knots in the standard contact 3-sphere, i.e. contact surgeries that yield again the standard contact 3-sphere. The main result is that we can exclude non-trivial cosmetic contact surgeries along all transverse knots not isotopic to the transverse unknot with self-lin…
Positive surgery on knots in weakly fillable 3-manifolds yields weakly fillable contact structures.
problem Conditions for a knot to admit a fillable positive surgery.
method Contact surgery, symplectic embeddings, and quasipositive knots.
result Conditions for a knot to admit a fillable positive surgery, including quasipositivity and slice genus equality.
Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.
problem Contact cosmetic surgeries for Legendrian knots in L-spaces.
method Adapting techniques from S3 to L-spaces, incorporating Heegaard Floer theory constraints.
result Contact cosmetic surgery conjecture holds for non-trivial Legendrian knots, except for Lagrangian slice knots.
In this note we show that +1-contact surgery on distinct Legendrian knots frequently produces contactomorphic manifolds. We also give examples where this happens for −1-contact surgery. As an amusing corollary we find overtwisted contact structures that contain a large number of distinct Legendrian knots with the s…
New contact Kirby moves complete the set for contact surgery diagrams.
problem Contact surgery diagrams and their relation to contactomorphic contact manifolds.
method Introducing lantern moves and chain moves to complete the set of contact Kirby moves.
result Two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of specific moves.
Extends LOSS invariant naturality to positive contact surgeries.
problem Naturality of LOSS invariant under positive contact surgeries.
method Proves naturality of LOSS invariant for Legendrian knots under positive integer contact surgeries.
result Contact invariant of +n surgery depends only on classical invariants of S.
Study contact structures on projective spaces, proving infinite non-isotopic structures.
problem Classify contact structures on projective spaces with specific surgery numbers.
method Detailed analysis of Gompf's Γ-invariant and d_3-invariant of tangential 2-plane fields.
result Infinitely many non-isotopic contact structures on real projective 3-space.
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.
Disproves conjectures about shared surgeries for distinct knots.
problem Knots sharing surgeries for multiple slopes.
method Constructing pairs of distinct knots with shared Dehn surgeries.
result Found pairs of distinct knots with four distinct shared slopes.
Study contact invariants using Floer homology to understand knots.
problem Understanding contact invariants of knots through surgeries.
method Used Heegaard Floer based contact invariants, sutured Floer homology, and limit constructions.
result Established relations between contact invariants of knots through surgeries.
By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this …
We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure ξK is supported by the fibred knot K⊂M, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer c…
The paper explores nonexistence and existence of symplectic and Stein fillable contact structures on 3-manifolds.
problem Nonexistence and existence of symplectic and Stein fillable contact structures on 3-manifolds.
method Construction of 3-manifolds and analysis of Dehn surgeries.
result The existence and nonexistence of fillable contact structures on specific 3-manifolds.
Study tight contact structures on figure-eight knot surgeries.
problem Classify tight contact structures on surgeries of figure-eight knot.
method Analyzes surgeries on figure-eight knot, determining tightness, symplectic fillability, and universality.
result First classification of tight contact structures on surgeries of figure-eight knot.
Classification of fillings for a specific type of contact surgery on the Hopf link.
problem Classifying the minimal symplectic fillings of a specific type of contact surgery on the Hopf link.
method Performed contact (-1)-surgery on a Legendrian representative of the Hopf link, classified the minimal symplectic fillings up to homeomorphism.
result Extended the classification of fillings to a broader range of structures, including those not universally tight.
Study shows patterns in Stein fillability of trefoil surgeries.
problem Determining Stein fillability of contact structures.
method Analyzing surgeries on the trefoil knot.
result Patterns in Stein fillability identified.
We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.
Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at l…
We use the Ozsváth-Szabó contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres −Σ(2,3,6n−1).
We show that all positive contact surgeries on every Legendrian figure-eight knot in (S3,ξstd) result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.
New techniques reveal tight contact manifolds with vanishing contact homology.
problem Understanding closed tight contact manifolds with vanishing contact homology.
method Developed algebraic tools and techniques to study holomorphic curves in surgery cobordisms.
result First known examples of closed tight contact manifolds with vanishing contact homology.
We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots - …