Cosmetic surgeries on pretzel knots are unique.
problem Understanding unique three-manifolds from surgeries on knots.
method Analyzing pretzel knots through Dehn surgeries.
result All pretzel knots satisfy the cosmetic surgery conjecture.
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On contact cosmetic surgerymath.GT
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 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.
Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.
problem Cosmetic surgeries on knots in homology spheres and their constraints.
method Rational surgery formula of the Casson-Walker invariant for 2-component links.
result Constraints on knots and surgery slopes for cosmetic surgeries.
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.
The study calculates and analyzes alternating surgeries for various knots.
problem Identifying and understanding alternating surgeries on knots.
method Algorithmic computation and structural analysis of alternating surgery slopes.
result The set of alternating surgery slopes is algorithmically computable and exhibits interesting phenomena.
New Heegaard Floer homology findings block chirally cosmetic surgeries.
problem Chirally cosmetic surgeries on knots and manifolds.
method Heegaard Floer homology, immersed curve formulations, and surgery formula.
result New obstructions to chirally cosmetic surgeries identified.
A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifer…
Quasi-alternating surgeriesmath.GT
The study examines quasi-alternating surgeries on knots and their properties.
problem Understanding quasi-alternating surgeries and their relation to L-space knots.
method Exploration of quasi-alternating surgeries, classification of knots, and analysis of slopes.
result All SnapPy census L-space knots except two admit quasi-alternating surgeries.
A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…
The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.
problem Proving non-existence of purely cosmetic surgeries for certain types of knots.
method Analyzing specific knot types with reduced diagrams and twist numbers, and using properties of Dehn surgeries.
result Large alternating Montesinos knots do not admit purely cosmetic surgeries.
New findings on cosmetic surgeries for satellite knots.
problem Cosmetic surgeries on knots and their properties.
method Analyzing satellite knots and their hyperbolic structures.
result Existence of hyperbolic satellite knots with cosmetic surgeries.
We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus kno…
Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.
problem Characterizing chirally cosmetic surgeries on specific knot types.
method Recent methods of Ichihara, Ito, and Saito applied to genus 2 and 3 alternating odd pretzel knots.
result Most genus 2 and 3 alternating odd pretzel knots do not admit chirally cosmetic surgeries.
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.
The purely cosmetic surgery conjecture is true for the Kinoshita-Terasaka and Conway knot familiesmath.GT
Two knot families meet cosmetic surgery conjecture.
problem Cosmetic surgery conjecture in knot theory.
method Analyzing Kinoshita-Terasaka and Conway knot families.
result All nontrivial members of both knot families satisfy the conjecture.
Study proves nontrivial knots can't undergo cosmetic surgeries.
problem Cosmetic surgeries on nontrivial knots.
method Calculated finite type invariant of order 3.
result Nontrivial knots do not admit chirally cosmetic surgeries.
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.
New insights into cosmetic surgeries using Heegaard Floer homology.
problem Understanding cosmetic surgeries on knots and three-manifolds.
method Using Heegaard Floer homology and knot Floer homology.
result Cosmetic surgeries on certain knots have specific coefficient constraints.
Infinite hyperbolic knots yield unusual surgeries.
problem Finding knots with specific surgery results.
method Examined infinite families of hyperbolic knots and their surgeries.
result Discovered knots with surgeries producing graph manifolds with five disjoint, non-parallel incompressible tori.
How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the ot…
By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery de…
Enhances knot surgery formulae for instanton Floer homology.
problem Calculating instanton Floer homology for various 3-manifolds.
method Integral surgery formulae and rational surgery formulae for null-homologous knots.
result Characterizes instanton Floer homology of specific 3-manifolds and knots.
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 on reducing surgeries on knots, developing thickness and genus bounds.
problem Understanding reducible surgeries on knots in .
method Developed thickness bounds for L-space knots and lower bounds on slice genus; used -invariants and mapping cone formula from Heegaard Floer homology.
result Provided new upper bounds on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots; verified the Cabling Conjecture for thin knots.
The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
problem Identifying lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
method Developed a lattice embedding obstruction to realize L-space surgeries on knots in the Poincaré homology sphere.
result Identified the only two knots in the Poincaré homology sphere that admit half-integer lens space surgeries.
Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery …
The article confirms two quasi-alternating surgeries for 9 asymmetric L-space knots.
problem Understanding quasi-alternating surgeries on asymmetric L-space knots.
method Using the Montesinos trick to confirm known surgeries.
result Confirmation of two quasi-alternating surgeries for each of 9 asymmetric L-space knots.
The paper proves a theorem about shared Dehn surgeries between knots.
problem Understanding shared Dehn surgeries between distinct knots.
method Uses the JSJ decomposition of knot exteriors to provide an effective bound.
result Provides an effective bound on the maximal number of shared surgeries between knots.
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.
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…
Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be cla…
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.
A Dehn surgery on a knot in is exceptional if it produces a reducible, toroidal or Seifert fibred manifold. It is known that a large arborescent knot admits no such surgery unless it is a type II arborescent knot. The main theorem of this paper shows that up to isotopy there are exactly three large arborescen…
Extends knot surgery to exotic four-manifolds.
problem Constructing exotic definite four-manifolds.
method Generalizes RBG construction to -surgery.
result Produces pairs of knots with same -surgery.
We show that if a Montesinos knot admits a Dehn surgery yielding a toroidal Seifert fibered 3-manifold, then the knot is the trefoil knot and the surgery slope is 0.
The study confirms that most positive 2-bridge knots up to 31 crossings do not have chirally cosmetic surgeries.
problem Determining which positive 2-bridge knots up to 31 crossings have chirally cosmetic surgeries.
method Using the obstruction formula from arXiv:2112.03144 and a Python program to compute classical knot invariants.
result Positive 2-bridge knots up to 31 crossings, except torus knots, do not admit chirally cosmetic surgeries.
We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.
Study of knot surgeries and JSJ decompositions to tackle -space conjecture.
problem Understanding JSJ decompositions and -space knots through Dehn surgeries.
method Slope detection techniques applied to toroidal 3-manifolds and rational surgeries.
result JSJ graphs of certain knot exteriors are rooted intervals, implying left-orderable fundamental groups.
New formula for dual knots using involutions.
problem Understanding dual knots and their transformations.
method Involutive analog of knot surgery formula.
result Computed local equivalence class for involutive dual knots.
The construction of knots via annular twisting has been used to create families of knots yielding the same manifold via Dehn surgery. Prior examples have all involved Dehn surgery where the surgery slope is an integral multiple of 2. In this note we prove that for any integer there exist infinitely many different k…
It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parame…
Exceptional Dehn surgeries have been classified for 2-bridge knots and Montesinos knots of length at least 4. In this paper we classify all toroidal Dehn surgeries on Montesinos knots of length 3.
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.
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.
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…
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.
Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic manifolds. For a null-homologous knot with certain conditions on the Thurston norm of the ambient manifold, if the knot admits cosmetic surgeries, then the surgery coefficients are equal up to sign.