Round surgery diagrams represent 3-manifolds in S3.
problem Representing and manipulating 3-manifolds in S3. method Introducing round surgery diagrams and defining moves to establish Kirby Calculus.
result Any 3-manifold can be obtained by a round surgery on a framed link in S3. Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
problem Understanding the generic properties of hyperbolic 3-manifolds.
method Counting model on links and Dehn surgeries.
result Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
3-manifolds can be created from braids.
problem Creating 3-manifolds from braids.
method Proving every 3-manifold is a Dehn surgery on a braid positive link.
result Every closed, oriented, connected 3-manifold is a Dehn surgery on a braid positive link.
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.
New surgeries found in 3D shapes without 2-spheres.
problem Finding non-hyperbolic surgeries in 3-manifolds.
method Combining work on hyperbolic knots and 3-manifolds with observations about surgeries.
result Every 3-manifold with certain properties contains a hyperbolic knot with a non-trivial surgery.
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.
3-manifolds can be made parallelisable via surgery.
problem Making 3-manifolds parallelisable.
method Surgery presentation, refined by Kaplan, contact geometry.
result All closed, orientable 3-manifolds are parallelisable.
Dehn surgery on knots in 3-manifolds rarely results in surface fibered manifolds.
problem Understanding when Dehn surgery on knots in 3-manifolds results in surface fibered manifolds.
method Analyzing the fiberedness of Dehn surgeries on knots in 3-manifolds.
result Dehn surgery on knots in 3-manifolds is not surface fibered for all but a sparse set of slopes.
Study calculates Reidemeister torsions for 3-manifolds from specific surgeries.
problem Calculating Reidemeister torsions for 3-manifolds from Dehn surgeries.
method Determined adjoint Reidemeister torsion for specific 3-manifolds.
result Reidemeister torsion vanishes for the studied 3-manifolds.
New examples of 3-manifolds not obtained by knot surgery found.
problem Finding 3-manifolds not obtainable by knot surgery.
method Provided two infinite families of examples of 3-manifolds.
result First examples of irreducible 3-manifolds with b1=1 not obtained by knot surgery.
The paper explores various surgery equivalence relations on 3-manifolds.
problem Understanding different equivalence relations on 3-manifolds.
method Restricting surgeries and using mapping class groups of surfaces.
result Characterization of surgery equivalence relations using invariants.
Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing. Massuyeau generalized this result to closed, spin 3-manifolds, and the second autho…
Extends Seifert algorithm to 3-manifolds via surgery.
problem Construct Seifert surfaces in arbitrary 3-manifolds.
method Uses surgery on framed links in S^3 to extend classical algorithm.
result Explicit construction of Seifert surfaces in 3-manifolds.
Study shows surgeries on certain knots yield left-orderable 3-manifolds.
problem Integral Dehn surgeries on genus one fibered knots in lens spaces.
method Classification of genus one fibered knots by Baker and analysis of surgeries.
result All integral Dehn surgeries on these knots yield left-orderable 3-manifolds.
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.
Study shows certain knots can be surgically altered to create non-left-orderable 3-manifolds.
problem Understanding non-left-orderable fundamental groups in 3-manifolds.
method Analyzing surgeries on negatively twisted torus knots.
result Certain knots can be surgically altered to create 3-manifolds with non-left-orderable fundamental groups.
Two-bridge knots can't be transformed into each other by certain surgeries.
problem Understanding transformations of 3-manifolds via knot surgeries.
method Used signature, finite type invariants for knots and SL(2,C) Casson invariant for 3-manifolds. result Two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries.
We study the spectrum of the Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities and its dependence on the generalized Dehn surgery coefficients.
Paper restricts chirally cosmetic surgeries on knots.
problem Chirally cosmetic surgeries on knots.
method Examined exceptional or half-integral chirally cosmetic surgeries.
result Obtained several restrictions on chirally cosmetic surgeries.
New proof for a knot type not admitting certain surgeries.
problem Chirally cosmetic surgeries on non-torus 2-bridge knots.
method Proof by contradiction and geometric analysis.
result Proves non-torus 2-bridge knots do not admit chirally cosmetic 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). 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.
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…
New examples of 3-manifolds not obtained by surgery on knots.
problem Identifying 3-manifolds not obtained by Dehn surgery.
method Combining Furuta's 10/8-theorem with combinatorial arguments.
result Found new 3-manifolds with weight one fundamental group not obtained by surgery.
Quantum algorithm finds Dehn filling slopes for hyperbolic 3-manifolds.
problem Finding exceptional Dehn filling slopes for hyperbolic 3-manifolds.
method Quantum 3D index using ideal triangulation of cusped 3-manifolds.
result Algorithm tested successfully on many examples.
We refine Matveev's result asserting that any two closed oriented 3-manifolds can be related by a sequence of borromean surgeries if and only if they have isomorphic first homology groups and linking pairings. Indeed, a borromean surgery induces a canonical isomorphism between the first homology groups of the involved …
The study limits the number of cosmetic surgeries for certain knots in specific 3-manifolds.
problem Limits the number of cosmetic surgeries for knots in specific 3-manifolds.
method Rational surgery formula for Casson--Walker--Lescop invariant, constraints for null-homologous knots.
result At most two pairs of integral purely cosmetic surgeries for null-homologous knots in rational homology spheres.
Study chirally cosmetic surgeries on knots with constraints and invariants.
problem Understanding chirally cosmetic surgeries on knots.
method Use original and SL(2,C) Casson invariants. result Complete classification of chirally cosmetic surgeries on genus one alternating knots.
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…
Computes Vafa-Witten invariants of 3-manifolds.
problem Computing invariants of 3-manifolds.
method Explicit computations using Vafa-Witten theory.
result Explicit invariants computed for specific 3-manifolds.
Study shows surgeries on certain knots result in non-left-orderable 3-manifold groups.
problem Understanding when surgeries on knots result in non-left-orderable fundamental groups.
method Analyzing fundamental groups of 3-manifolds obtained by surgeries on specific knots.
result Fundamental groups are not left-orderable under certain conditions on surgery parameters.
Study the JSJ-decomposition of a specific 3-manifold.
problem Classify the JSJ-decomposition of a 3-manifold from 0-surgery on a pretzel knot.
method Utilize the classification of exceptional fillings of minimally twisted five-chain links.
result Determine the JSJ-decomposition of the 3-manifold.
The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.
problem Checking cosmetic surgeries on knots and 3-manifolds.
method Knot invariants and hyperbolic geometry.
result Verification of no cosmetic surgeries on knots and 3-manifolds.
3-braid knots can't have purely cosmetic surgeries.
problem Cosmetic surgeries on 3-braid knots.
method Using recent work on knot closures, proved no purely cosmetic surgeries exist.
result 3-braid knots do not admit purely cosmetic surgeries.
Alexander polynomial constraints derived from Seifert surgeries.
problem Constraints on knot surgery results.
method Dijkgraaf-Witten invariant applied to abelian groups.
result Alexander polynomial constraints on knot surgeries.
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.
We determine the Dehn surgeries on 2-bridge links, which yield reducible 3-manifolds. Further, we show the conditions that we obtain a torus or cable knot from one component of a 2-bridge link by a surgery on another component.
In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery on geometrically finite hypebolic cone-manifolds.
New method finds infinitely many knots in 3-manifolds.
problem Finding distinct knots in 3-manifolds.
method Producing infinitely many pairs of distinct knots with homeomorphic surgeries.
result Found infinitely many distinct knots with homeomorphic surgeries.
We study Dehn surgeries on null-homotopic knots that yield fibred 3--manifolds when an additional (but natural) homological restriction is imposed. The major tool used is Gabai's theory of sutured manifold decomposition. Such surgeries are negative examples to a question of Michel Boileau. Another result we will prov…
Proves volume conjectures for figure-eight knot surgeries.
problem Volume conjectures for hyperbolic 3-manifolds.
method Ohtsuki's method applied to figure-eight knot surgeries.
result Proves Asymptotic Expansion and Volume Conjectures for figure-eight knot surgeries.
In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on (S3,ξstd) along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an inf…
Unique surgery descriptions found for knots in 3-manifolds.
problem Characterizing and understanding unique surgery descriptions for knots in 3-manifolds.
method Analyzing infinitely many surgeries along knots and hyperbolic L-space knots, proving unique descriptions.
result Infinitely many surgeries along knots have unique descriptions, generalizing the concept of characterizing slopes.
New method to calculate 3-manifold invariants via skew-racks.
problem Calculating invariants of 3-manifolds.
method Introducing skew-racks with good involution and Property FR, defining cocycle invariants.
result Established new approach to obtain 3-manifold invariants via Dehn surgery.
We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expresse…
We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally …
For a hyperbolic knot in the 3-sphere, at most finitely many Dehn surgeries yield non-hyperbolic 3-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed 3-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal s…
The study finds generalized torsion elements in 3-manifolds from specific knot surgeries.
problem Identifying generalized torsion elements in 3-manifolds from knot surgeries.
method Using the JSJ-decomposition of the 3-manifold and bi-orderability properties.
result Existence and properties of generalized torsion elements in specific 3-manifolds.