Paper shows how to transform certain flows into R-covered ones.
problem Transforming Anosov flows into R-covered ones.
method Goodman-Fried surgery along periodic points.
result Any topologically transitive Anosov flow can be transformed into an R-covered one.
Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.
problem Understanding and manipulating pseudo-Anosov flows.
method Performing horizontal surgery on pseudo-Anosov flows by cutting along specific annuli and regluing with a Dehn twist.
result Horizontal Goodman surgery on transitive pseudo-Anosov flows yields an almost equivalent flow.
Shows Anosov flows with genus one sections, supporting a conjecture.
problem Finding genus one Birkhoff sections for Anosov flows.
method Utilizes horizontal Goodman surgery operation and correspondence with veering triangulations.
result Provides evidence for Fried and Ghys conjecture.
Generalizes surgery techniques for projectively Anosov flows.
problem Creating new projectively Anosov flows from existing ones.
method Introducing a generalized Goodman surgery technique for projectively Anosov flows.
result Generates new examples of projectively Anosov flows on hyperbolic 3-manifolds.
Classifies Anosov flows on figure-eight knot surgeries.
problem Classifying Anosov flows on Dehn surgeries on the figure-eight knot.
method Combining Plante's classification of M(0) with Schwider's branched surfaces.
result M(r) carries a unique Anosov flow if r is an integer, and no Anosov flow if r is not an integer.
New surgery method preserves Anosov flow properties using bi-contact geometry.
problem Defining a new type of surgery on Anosov flows.
method Bi-contact geometry and Reeb dynamics.
result Necessary and sufficient condition for generating contact Anosov flows.
This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to filli…
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.
Legendrian arcs connect veering triangulations to Anosov flows.
problem Connecting veering triangulations to Anosov flows for study.
method Realizing edges as Legendrian arcs with a bicontact structure.
result Veering triangulations can be placed in steady position.
Unified solution to Goodman-Pollack transversal problem using matroids and topology.
problem Existence of an affine k-dimensional transversal to convex sets.
method Matroidal joins and topological methods.
result Unified solution including colorful Helly theorem and Holmsen's theorem.
Torus leaves play a crucial role in the theory of foliations. For example non-taut foliations admit a torus leaf (see the article of Goodman). In this paper, we study all the foliations near a torus leaf, and try to understand why sometimes it is taut, or non-taut (and Reebless). We focus on some crucial examples to un…
We construct 2^{Ω(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3-spheres …
We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local version of Harer's twisting, which is presented in two different (but stably equivale…
We study open books (or open book decompositions) of a closed oriented 3-manifold which support overtwisted contact structures. We focus on a simple closed curve along which one can perform Stallings twist, called ``twisting loop''. We show that the existence of a twisting loop on the fiber surface of an open book is e…
Stochastic gradient descent outperforms traditional force-directed methods.
problem Improving graph layout quality and efficiency.
method Applying stochastic gradient descent for stress minimization.
result Stochastic gradient descent is simpler and more robust than traditional methods.
This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this…
When response variables are nominal and populations are cross-classified with respect to multiple polytomies, questions often arise about the degree of association of the responses with explanatory variables. When populations are known, we introduce a nominal association vector and matrix to evaluate the dependence of …
Recurrent correlation neural networks (RCNNs), introduced by Chiueh and Goodman as an improved version of the bipolar correlation-based Hopfield neural network, can be used to implement high-capacity associative memories. In this paper, we extend the bipolar RCNNs for processing hypercomplex-valued data. Precisely, we …
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 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…
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…
Quantum invariants for fibered links determined by genus and Hopf invariant.
problem Quantum invariants of fibered links in S3. method Genus bounds and Giroux-Goodman theorem on fiber surfaces.
result Top coefficient of ADO invariant is determined by Hopf invariant.
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.
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.