Surgery paths in sphere graph have limited distance.
problem Understanding distances between surgery paths in the sphere graph.
method Relied on understanding surgeries' effect on Guirardel core and equivalence to Rips moves.
result Hausdorff distance between any two surgery paths is at most 4.
A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pair…
Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's spher…
Proves path-connectedness of metrics on 3-manifolds with positive scalar curvature.
problem Proving path-connectedness of metrics on 3-manifolds with positive scalar curvature.
method Uses Ricci flow with surgery and infinite connected sums with geometry control.
result Proves the moduli space of metrics is path-connected.
The paper extends symplectic techniques to generalized complex geometry.
problem Creating stable generalized complex structures on high-dimensional manifolds.
method Introducing generalized Luttinger surgery and generalized Gluck twist.
result Produced stable generalized complex structures with non-homotopy-equivalent components.
The paper constructs paths of SL2(R) representations for pretzel knots and shows left-orderability conditions.
problem Left-orderability of knot surgery manifolds.
method Explicit construction of continuous paths of SL2(R) representations.
result Fundamental groups of certain knot surgeries are left-orderable.
Extends metric properties over surgeries to higher codimensions.
problem Extending metrics of positive scalar curvature over surgeries in higher codimensions.
method Generalizes Gromov-Lawson, Schoen-Yau, and Walsh's construction to (p,n)-intermediate scalar curvature. result Shows infinitely many path components for certain metrics on (4n−1)-manifolds. We give estimates on the length of paths defined in the sphere model of outer space using a surgery process, and show that they make definite progress in some sense when they remain in some thick part of outer space. To do so, we relate the Lipschitz metric on outer space to a notion of intersection numbers.
We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifold…
Study shows path-connectedness of tori moduli space for n≥3 and knot class correspondence for n=2.
problem Topology of moduli space of two-convex embedded tori.
method Variant of mean curvature flow with surgery, preserving topology, embeddedness, and two-convexity.
result Proves path-connectedness for n≥3 and knot class correspondence for n=2.
The study finds infinitely many periodic orbits that can be used to modify Anosov flows.
problem Can surgeries on periodic orbits of Anosov flows produce equivalent flows?
method Analyzing suspension Anosov flows, the study identifies pairs of periodic orbits that can be used to modify the flow.
result For some suspension Anosov flows, there exist infinitely many pairs of periodic orbits that can be used to modify the flow.
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…
Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.
problem Finding homology 3-spheres with non-Morse-Bott Chern-Simons functions.
method Constructing specific surgeries on torus knots.
result Examples of homology 3-spheres with non-Morse-Bott Chern-Simons functions.
We apply Lescop's construction of Z-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of "AL-paths" given in arXiv:1403.8030. We obtain an invariant Z^n of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over S1, whi…
Machine learning assesses surgical skill in robotic-assisted surgery.
problem Subjective evaluation of surgeon skill in robotic-assisted surgery.
method Machine learning applied to six movement features for classification.
result Framework classifies surgical skill with 85.7% accuracy.
We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold Pm with normal bundle $\oplus_{j=1}^{n-m}\mathcal{O}_…
Study on pretzel knots and their left-orderable properties.
problem Investigating left-orderability of 3-manifolds derived from pretzel knots.
method Continuous paths of elliptic SL2(R) representations.
result Left-orderable fundamental groups for specific surgeries and branched covers.
The paper describes distances on Sol-type groups using novel geometric techniques.
problem Understanding distances on Sol-type groups.
method New technique of Euclidean curve surgery to describe uniformly roughly geodesic paths.
result The rough isometry type of distances on Sol-type groups is determined by a specific metric restriction.
This paper restricts efficient geodesics to non-separating curves.
problem Finding efficient geodesics in the complex of curves.
method Analysis of the dot graph and surgeries.
result Efficient geodesics can be restricted to the non-separating curve complex.
Study on metrics with positive scalar curvature and convex boundary.
problem Characterizing 3-manifolds with positive scalar curvature and convex boundary.
method Combination of earlier contributions, smoothing procedure, Ricci flow, and conformal deformation techniques.
result Path-connectedness of the moduli space of metrics.
In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path c…
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.
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.
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 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.
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between…
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.
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.
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. We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.
Study links with specific surgeries in 4-manifolds.
problem Understanding which links have surgeries yielding specific 4-manifolds.
method Examining trisections and link surgeries in 4-manifolds.
result Found families of links with the specified surgeries, but not all.
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 finds chirally cosmetic surgeries on knots and manifolds, contradicting previous conjectures.
problem Disprove conjectures about chirally cosmetic surgeries and purely cosmetic surgeries.
method Construct infinite families of chirally cosmetic surgeries and purely cosmetic surgeries on specific geometric objects.
result Found chirally cosmetic surgeries and purely cosmetic surgeries that contradict previous conjectures.
New surgery operation preserves monotonicity of Lagrangians.
problem Preserving monotonicity of Lagrangians in surgery operations.
method BSP surgery, wall-crossing formula for disk-potentials.
result BSP surgery can preserve monotonicity of Lagrangians.
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.