The paper generalizes trisection definitions and applies them to specific cases.
problem Defining and applying trisections to 4-manifolds and surfaces.
method Generalized trisection definitions and their application to specific cases.
result Classification of an infinite family of genus three trisections.
Analytic surgery and gluing formula for torsion forms in fiber bundles.
problem Behavior of torsion forms under analytic surgery in fiber bundles.
method Analytic surgery and gluing formula for Bismut-Lott torsion and eta forms.
result Gluing formula for Bismut-Lott analytic torsion and eta forms under surgery limit.
Let Θ(M,K) denote the 2-loop piece of (the logarithm of) the LMO invariant of a knot K in M, a ZHS^3. Forgetting the knot (by which we mean setting diagrams with legs to zero) specialises Θ(M,K) to λ(M), Casson's invariant. This note describes an extension of Casson's surgery formula for his invariant to Θ(M,K). To be …
Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
problem Verifying the asymptotic expansion conjecture for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
method Using logarithmic holonomies of meridians and hyperbolic cone structures, the study verifies the conjecture for pairs where M∖L is homeomorphic to a fundamental shadow link complement. result The asymptotic expansion conjecture is true for pairs (M,L) with sufficiently small cone angles and M∖L homeomorphic to a fundamental shadow link complement. In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.
We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any re…
We introduce blow-up and blow-down operations for generalized complex 4-manifolds. Combining these with a surgery analogous to the logarithmic transform, we then construct generalized complex structures on nCP2 # m \bar{CP2} for n odd, a family of 4-manifolds which admit neither complex nor symplectic structures unless…
We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits…
The Whitehead link complement's geometry and topology are studied using ideal triangulations and tropical geometry.
problem Understanding the geometry and topology of the Whitehead link complement and its Dehn surgeries.
method Ideal triangulations, spun-normal surfaces, and tropical geometry.
result All boundary curves of the Whitehead link complement are strongly detected by its character variety.
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.
Round handles are affiliated with smooth 4-manifolds in two major ways: 5-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas 4-dimensional round handles are the building blocks of broken Lefschetz fibrations on them. The purpose of this article is to shed more…
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.
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.
We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.
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…
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 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.
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.
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.
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 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 study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the SL(2,C) version of Casson invariant…
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.
Extends knot surgery to exotic four-manifolds.
problem Constructing exotic definite four-manifolds.
method Generalizes RBG construction to n-surgery. result Produces pairs of knots with same n-surgery. Study exact surgery formula in involutive Heegaard Floer homology.
problem Understanding integer homology spheres through knot surgery.
method Using doubling model of involution and mapping cone formula.
result Examples of non-homology cobordant integer homology spheres.
The paper develops surgery theories for foliations and solves a problem posed by Weinberger.
problem Classifying leaves of foliations and verifying Borel conjectures.
method Developed two surgery theories and two kinds of Whitehead torsion for foliations.
result Solved a problem posed by Weinberger and verified conjectures in some cases.
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.
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…
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.
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…
In this paper, we use Heegaard Floer homology to study reducible surgeries. In particular, suppose K is a non-cable knot in the three-sphere with an L-space surgery. If p-surgery on K is reducible, we show that p equals 2g(K)-1. This implies that any knot with an L-space surgery has at most one reducible surgery, a fac…
Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.
problem Mean curvature flow with surgery for compact mean convex hypersurfaces.
method Topological surgeries performed by the flow itself through nondegenerate cylindrical singularities, adjusted at smooth times.
result Extends previous results for 2-convex flows and constructs a flow for compact mean convex hypersurfaces.
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.
This paper constructs Koszul duals for Heegaard Floer Dehn surgery formulas.
problem Computational problems involving link surgery formulas.
method Described Koszul duals of an associative algebra encoding Heegaard Floer Dehn surgery formulas.
result Proved several equivalences of categories using dualizing bimodules.