The LMO invariant helps find constraints for cosmetic surgeries on knots.
problem Finding constraints for cosmetic surgeries on knots.
method Using the LMO invariant to analyze surgeries on knots.
result Constraints for knots to admit cosmetic surgeries and Lens space surgeries.
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.
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.
New proof of Alexander polynomial constraints for lens space surgeries.
problem Constraints on Alexander polynomials for lens space surgeries.
method Using changemaker lattices to prove a theorem.
result Constraints on Alexander polynomials for specific 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.
Study constraints on knot surgery invariants using Seiberg-Witten theory.
problem Understanding invariants of knots in the three-sphere.
method Use Manolescu correction terms and Seiberg-Witten theory.
result Constraints on invariants for knots in the three-sphere.
Study shows most knots up to 10 crossings can't be chirally cosmetic.
problem Identifying knots that can undergo chirally cosmetic surgeries.
method Used invariants from 3-manifold theory, including quantum SO(3)-invariant and Heegaard Floer homology.
result Approximately 75% of knots up to 10 crossings do not admit chirally cosmetic surgeries.
We solve a conjecture about surgeries on special 3D shapes.
problem Cosmetic surgery conjecture in 3-manifold theory.
method Proved constraints on exceptional surgeries for homology spheres.
result At most one pair of exceptional truly cosmetic slopes for non-trivial surgeries.
Geometric surgery theory applied to quantum statistics in spacetime.
problem Constraints of quantum statistics data in spacetime dimensions.
method Geometric-topology surgery theory applied to spacetime manifolds.
result New quantum surgery constraints derived, analogous to Verlinde formula.
In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.
We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsváth and Szabó's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on p…
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.
Derives the Space-Time Positive Mass theorem in arbitrary dimensions.
problem Proving the Space-Time Positive Mass theorem in arbitrary dimensions.
method Skin structures and surgeries on minimal and marginally outer trapped hypersurfaces.
result Derives the Space-Time Positive Mass theorem in arbitrary dimensions.
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.
Quantum surgery formulas link statistics and topology.
problem Formulate constraints for quantum statistics of entangled systems.
method Geometric-topology surgery theory on spacetime manifolds.
result New quantum surgery formulas and constraints derived.
Smith inequality proven for 3-manifold covers in Floer homology.
problem Understanding regular covers of 3-manifolds in Floer homology.
method Monopole Floer homology and Heegaard Floer correspondence.
result 3-manifolds with certain cover properties are L-spaces.
In an earlier paper, we used the absolute grading on Heegaard Floer homology to give restrictions on knots in S3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the non-zero coefficients …
We derive the Riemannian Positive Mass theorem in arbitrary dimensions, without any topological constraints. The main new tools are skin structures and surgeries on minimal hypersurfaces.
New method for scalar curvature studies without manifold restrictions.
problem Scalar curvature constraints without manifold restrictions.
method Skin structures merging minimal hypersurface methods with surgery arguments.
result Obstruction and structure theories derived for scalar curvature constraints.
Suspensions of manifolds by circle surgeries are key in free action constructions.
problem Understanding free S1-actions on smooth manifolds of dimension at least 3. method Circle surgeries on S1imesM yield suspensions Σ0M and Σ1M. result Suspension operations Σi are fundamental in constructing and classifying manifolds with free S1-actions. New research shows that many slopes are characterizing for satellite knots.
problem Characterizing slopes for satellite knots.
method Detailed examination of the JSJ decomposition of a surgery along a knot, combined with other authors' constraints on surgery slopes.
result Many non-integral slopes are characterizing for composite knots.
Study shows constraints on cosmetic surgeries in S3 for knots with specific properties.
problem Cosmetic surgeries in S3 for knots with certain properties. method Use of Heegaard Floer homology and surgery formula.
result Explicit constraints on slopes for cosmetic surgeries, including ±2 and ±1/q. Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d and dˉ…
New geometric structures on 3-manifolds discovered and proven for all closed orientable ones.
problem Existence of specific geometric structures on 3-manifolds.
method Generalized geometry and surgery techniques.
result Any closed orientable 3-manifold admits a B3-generalized complex structure. New constraints found for Lagrangian embeddings in symplectic fillings.
problem Understanding Lagrangian embeddings in symplectic fillings with semisimple cohomology.
method Deriving constraints through Lagrangian embeddings and symplectic cohomology analysis.
result Existence of many non-toric monotone symplectic manifolds with proper wrapped Fukaya categories.
The paper calculates involutive Heegaard Floer homology for specific 3-manifolds.
problem Calculating numerical invariants for specific 3-manifolds.
method Involutive Heegaard Floer homology techniques and spin filling constraints.
result Established new constraints and obstructions for 3-manifolds.
Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying $rank(T) =…
If a knot K bounds a genus one Seifert surface F in the 3-sphere and F contains an essential simple closed curve alpha that has induced framing 0 and is smoothly slice, then K is smoothly slice. Conjecturally, the converse holds. It is known that if K is slice, then there are strong constraints on the algebraic concord…
Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either ±1 or 0 and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's p…
In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact 3-manifold is path-connected. The proof uses the Ricci flow with surgery, the conformal method, and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundam…
Study fermionic theories, their anomalies, and modular transformations.
problem Understanding fermionic theories and their anomalies.
method Use spin-cobordisms, surgeries, and invertible topological quantum field theories.
result Explicit combinatorial expressions for spin-bordism invariants.
Formula connects surgeries to Seiberg-Witten invariants.
problem Understanding how surgeries affect Seiberg-Witten invariants.
method Proves surgery formulas for Seiberg-Witten invariants and families.
result Expresses new invariants in terms of original ones.
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…
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.
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.
Study pochette surgery on 4-manifolds, focusing on 4-spheres.
problem Understanding pochette surgery on 4-spheres and its effects.
method Using linking number of pochette embeddings, compute homology and analyze surgeries.
result Pochette surgery on any homology 4-sphere can be computed via homology, and trivial cord surgeries do not change diffeomorphism type.
New method proves cosmetic surgery conjecture for certain knots.
problem Proving the cosmetic surgery conjecture for specific knots.
method Using filtered instanton homology and Chern-Simons filtration.
result Reduced the conjecture to surgeries of ±2 on genus 2 knots. 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.
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. 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.
The paper constructs homotopy 4-spheres using pochette surgery.
problem Creating homotopy 4-spheres from pochette surgeries.
method Pochette surgery generalizes Gluck surgery to construct embeddings of pochettes into the 4-sphere and proves homotopy 4-spheres are diffeomorphic to the 4-sphere.
result Homotopy 4-spheres obtained from pochette surgeries are all diffeomorphic to the 4-sphere.
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). This paper concerns the truly or purely cosmetic surgery conjecture. We give a survey on exceptional surgeries and cosmetic surgeries. We prove that the slope of an exceptional truly cosmetic surgery on a hyperbolic knot in S3 must be ±1 and the surgery must be toroidal but not Seifert fibred. As consequence we…
Complete exceptional surgeries identified for two-bridge links.
problem Identifying complete exceptional surgeries on two-bridge links.
method Enumerated all hyperbolic two-bridge links with complete exceptional surgeries.
result Listed all candidate slopes for complete exceptional surgeries.
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.
Study classifies exceptional surgeries on specific links.
problem Classifying exceptional Dehn surgeries on specific links.
method Examined minimally twisted seven-chain links.
result Classified all exceptional Dehn surgeries on the links.
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.
New surgery problems solved for 4D topological surgery.
problem Intermediate 4D surgery problems for 1/2-π1-null groups. method Group-theoretic 2-Engel relation and geometric applications.
result 1/2-π1-null surgery problems are universal.