The paper finds hyperbolic small knots in many 3-manifolds.
problem Finding small knots in 3-manifolds.
method Explicit examples of hyperbolic small knots in spherical 3-manifolds.
result Explicit examples of hyperbolic small knots in most spherical 3-manifolds.
New 4-manifolds without 1- and 3-handles are created from knots.
problem Creating 4-manifolds without specific handles. method Knot surgery with specific knots.
result Knot surgery 4-manifolds E(n)K have a handle decomposition without 1- and 3-handles. We show that if a surgery on a knot in a product sutured manifold yields the same product sutured manifold, then this knot is a 0-- or 1--crossing knot. The proof uses techniques from sutured manifold theory.
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
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.
Paper classifies CFK∞ type of almost L-space knots.
problem Understanding almost L-space knots and their properties.
method Utilizes Heegaard Floer homology and knot Floer homology.
result Classifies CFK∞ type of almost L-space 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 show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; f…
Null-homotopic knots in certain 3-manifolds are uniquely identified by their complements.
problem Identifying null-homotopic knots in specific 3-manifolds.
method Instanton Floer homology and SU(2)-representation varieties.
result Null-homotopic knots are determined by their complements in certain 3-manifolds.
Algorithm finds knot friends in 3-manifolds.
problem Identifying knots that share similar 3-manifold complements.
method Developed an algorithm using SnapPy and Regina.
result Constructed a census of simple knots with friends.
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.
We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. More…
The paper constructs exotic knotted surfaces and curves in 4-manifolds.
problem Understanding exotic knotted surfaces and curves in 4-manifolds.
method Local knotting and construction of surfaces and curves.
result First examples of exotically knotted complex curves and symplectic 2-spheres.
We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. …
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.
Research on knots and their 4-manifold covers.
problem Determining if knots are slice.
method Analyzing double branched covers and computer searches.
result Methods to decide if knots are slice.
We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation C(2n,3). We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
Kashaev's invariants for a knot in a three sphere are generalized to invariants of a knot in a three manifold. A relation between the newly constructed invariants and the hyperbolic volume of the knot complement is observed for some knots in lens spaces.
Lickorish has constructed large families of contractible 4--manifolds that have knotted embeddings in the 4--sphere and has also shown that every finitely presented perfect group with balanced presentation occurs as the fundamental group of the complement of a knotted contractible manifold. Here we make a few observati…
The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.
problem Checking cosmetic surgeries on knots and 3-manifolds.
method Knot invariants and hyperbolic geometry.
result Verification of no cosmetic surgeries on knots and 3-manifolds.
The study explores knots and manifolds, proving properties and non-left-orderable groups.
problem Properties of knots and manifolds after surgeries and fillings.
method Fixed point method and Dehn surgery.
result Fundamental groups of certain manifolds are not left orderable.
The paper confirms a conjecture about knots in aspherical 3-manifolds.
problem The study of topological concordance of knots in aspherical 3-manifolds.
method The method involves extending Milnor's link invariants to non-simply-connected 3-manifolds and employs computations.
result The paper confirms the conjecture for a large family of open cases, maximizing the number of almost-concordance classes.
Lower bound on crossing number of knots and links on a 2-sided surface in a 3-manifold
problem Crossing number of knots and links on a 2-sided surface in a 3-manifold
method Rank of fundamental groups
result Lower bound on crossing number
The paper extends knot theory to 4-manifolds, defining new genera and obstructions.
problem Understanding embeddings of 3-manifolds into 4-manifolds with specific properties.
method New L2-signature obstructions and extensions of knot genera. result Existence of knots with arbitrarily large generalized superslice genera and double stabilizing numbers.
Study on knots in contact manifolds, focusing on their width and thickness.
problem Understanding the width and thickness of knots in contact manifolds.
method Analyzing solid tori, defining width, and comparing it to Thurston-Bennequin invariant.
result Existence of non-thickenable tori in various knot types.
Constructs knots from 3-manifolds with specified geometric limits.
problem Build knots from 3-manifolds with specific geometric properties.
method Constructs explicit families of knots converging to specified 3-manifolds.
result Constructs knots from geometrically finite 3-manifolds with one end.
Study on fibered knots in 3-manifolds, proving unrelated volume and genus.
problem Unrelated volume and genus in hyperbolic fibered knots.
method Proof of unrelated volume and genus for hyperbolic fibered knots in 3-manifolds.
result Proved that volume and genus are unrelated for hyperbolic fibered knots.
New Legendrian bounds for non-fibered knots in 3-manifolds.
problem Understanding Legendrian representatives of non-fibered knots.
method Analyzing Thurston-Bennequin bounds and contact invariants.
result Non-fibered knots have Legendrian representatives with tb=0. New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.
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.
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.
The paper characterizes when a 2-sphere can be embedded in a knot trace.
problem Characterizing when a 2-sphere can be embedded in a knot trace.
method Using classical and computable knot invariants, the paper provides conditions for embedding a 2-sphere in a knot trace.
result Conditions for a knot to be topologically n-shake slice. Study calculates Reidemeister torsions for 3-manifolds from specific surgeries.
problem Calculating Reidemeister torsions for 3-manifolds from Dehn surgeries.
method Determined adjoint Reidemeister torsion for specific 3-manifolds.
result Reidemeister torsion vanishes for the studied 3-manifolds.
Lecture notes on Heegaard Floer homology for 3-manifolds and knots.
problem Understanding Heegaard Floer homology for 3-manifolds and knots.
method Explains Heegaard diagrams, Heegaard Floer homology, and knot Floer homology.
result Relationship between knot and 3-manifold invariants.
We prove a neighbourhood theorem for arbitrary knots in contact 3-manifolds. As an application we show that two topologically isotopic Legendrian knots in a contact 3-manifold become Legendrian isotopic after suitable stabilisations.
Researchers compute knot invariants in Seifert manifolds using Wilson loops.
problem Calculating BPS invariants for knots in Seifert manifolds.
method Analytically continued Chern-Simons level and representation from Wilson loop operator.
result Homological blocks for knots in Seifert manifolds and Seifert integer homology spheres.
We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We pr…
Survey on knot theory's impact on four-dimensional topology.
problem Understanding invariants of smooth four-manifolds.
method Using Kirby diagrams and Heegaard Floer theory.
result Progress in detecting exotic structures.
2-knot manifolds have Seifert fibered base orbifolds.
problem Characterizing Seifert fibered 2-knot manifolds.
method Analyzing aspherical 2-orbifolds with orbifold fundamental groups of weight 1.
result Aspherical 2-orbifolds with orbifold fundamental groups of weight 1 are Seifert fibered bases of 2-knot manifolds.
Study equivariant 4-genus of knots in symmetric 4-manifolds.
problem Understanding equivariant 4-genus of knots in symmetric 4-manifolds.
method Developed techniques for constructing slice disks via equivariant tubing construction.
result Equivariant 4-genus can differ from standard and equivariant 4-genus of 4-manifolds.
A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.
Study handles in sutured manifolds and knots, finding varied handle numbers and unique surfaces.
problem Understanding handle numbers and incompressible Seifert surfaces in sutured manifolds and nearly fibered knots.
method Extending Haken's Theorem, analyzing product annuli and disks, and examining specific knot types.
result Variety of handle numbers and unique incompressible Seifert surfaces in nearly fibered knots.
Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in S3. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if K is a knot in a closed 3--manifold Y, Y−K is irreducible, and HFK^(Y,K) is monic, then K is fibred. The proof relies …
Study proves non-left-orderability of 3-manifolds derived from specific knots.
problem Proving non-left-orderability of knot groups in 3-manifolds.
method Analyzing fundamental groups of cyclic branched covers of pretzel knots.
result Non-left-orderability of fundamental groups of n-fold cyclic branched covers of P(3,−3,−2k−1) for all integers k and n≥1. Let K be a rationally null-homologous knot in a three-manifold Y. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homology of rational surgerie…
Dehn surgery on knots in 3-manifolds rarely results in surface fibered manifolds.
problem Understanding when Dehn surgery on knots in 3-manifolds results in surface fibered manifolds.
method Analyzing the fiberedness of Dehn surgeries on knots in 3-manifolds.
result Dehn surgery on knots in 3-manifolds is not surface fibered for all but a sparse set of slopes.