The JSJ decomposition helps classify genus two handlebody-knots.
problem Classifying genus two handlebody-knots.
method JSJ decomposition and annulus classification applied to handlebody-knot exteriors.
result Genus two handlebody-knots classified using annulus diagrams.
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
Study confirms conjectures for topologically slice knots' concordance group.
problem Primary decomposition conjectures for knot concordance groups.
method Use of amenable L2-signatures, Ozsváth-Szabó d-invariants, and Némethi's Heegaard Floer homology results. result Smooth concordance group of topologically slice knots has large subgroup with true primary decomposition.
New findings on knot concordance show limitations to primary decompositions.
problem Understanding the structure of knot concordance groups.
method Analyzing homomorphisms and polynomial factorizations of Alexander polynomials.
result Primary decompositions of topologically slice knots cannot exist.
Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.
problem Understanding handle decompositions of knots surgered elliptic surfaces.
method Using Kirby diagrams derived from Lefschetz fibrations.
result Knot surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.
We construct an infinite collection of knots with the property that any knot in this family has n-string essential tangle decompositions for arbitrarily high n.
Flattenings of knotted surfaces help define new invariants.
problem Understanding and quantifying knotted surfaces in 4-sphere.
method Using hyperbolic decompositions and projections onto 2-sphere.
result Introduced layering, trunk, and partition number invariants.
Study the JSJ-decomposition of a specific 3-manifold.
problem Classify the JSJ-decomposition of a 3-manifold from 0-surgery on a pretzel knot.
method Utilize the classification of exceptional fillings of minimally twisted five-chain links.
result Determine the JSJ-decomposition of the 3-manifold.
Characterizes Legendrian knots in lens spaces.
problem Classifying Legendrian knots in lens spaces.
method Splitting lens spaces and using convex Heegaard decomposition.
result All Legendrian torus knots in universally tight lens spaces are classified.
We show that any non-minimal bridge decomposition of a torus knot is stabilized and that n-bridge decompositions of a torus knot are unique for any integer n. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…
This paper studies periodic and free periodic knots in alternating projections.
problem Understanding periodic and free periodic knots in alternating projections.
method Analyzing the essential Conway decomposition and Murasugi decomposition of alternating knots.
result Conditions for an alternating knot to be freely periodic are identified.
New insights into knot fusion numbers via cabling.
problem Understanding fusion numbers of ribbon knots and their behavior under cabling.
method Utilizing knot Floer homology and cabling formulas to analyze fusion numbers.
result The fusion number and strong homotopy fusion number of (p,1)-cable knots are preserved.
Enhanced Euler characteristic improves knot homology detection.
problem Detecting knots in instanton homology.
method Decomposing sutured instanton homology with torsion.
result Enhanced Euler characteristic equals SFH Euler characteristic.
In the present paper, we will show that for any integer n>0 there are infinitely many twisted torus knots with n-string essential tangle decompositions.
The Upsilon invariant helps classify fibered knots and their open book decompositions.
problem Classifying fibered knots and their open book decompositions.
method Using the Ozsváth-Stipsicz-Szabó concordance invariant Upsilon.
result Fibered knots satisfying a specific condition are either unique in their smooth concordance classes or provide counterexamples to the Slice-Ribbon Conjecture.
We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-widt…
We determine the structure of the circular handle decompositions of the family of free genus one knots. Namely, if k is a free genus one knot, then the handle number h(k)= 0, 1 or 2, and, if k is not fibered (that is, if h(k)>0), then k is almost fibered. For this, we develop practical techniques to construct circular …
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. M. Scharlemann has recently proved that any genus one tunnel number one knot is either a satellite or 2-bridge knot, as conjectured by H. Goda and M. Teragaito; all such knots admit a (1,1) decomposition. In this paper we give a classification of the family of (1,1) knots in S3 with crosscap number two (i.e., boundi…
The study shows conditions for elliptic surfaces without 1-handles.
problem Finding conditions for elliptic surfaces without 1-handles.
method Analyzing knot surgeries and log-transformations of elliptic surfaces.
result Conditions for elliptic surfaces without 1-handles are established.
Fundamental quandle helps in understanding knots and links.
problem Understanding the structure of knots and links.
method Using quandle categories and tangle decompositions.
result Derived presentations of fundamental quandles for various knots and links.
Proof confirms volume conjecture for a specific knot.
problem Verifying the volume conjecture for a specific knot.
method Using a generalized topological quantum field theory and a tetrahedral decomposition.
result Volume conjecture holds for the 73 knot in S3. Let K be a tunnel number two knot. Then, by considering the (g,b)-decompositions, K is one of (3, 0)-, (2, 1)-, (1, 2)- or (0, 3)-knots. In the present paper, we analyze the connected sum summands of composite tunnel number two knots and give a complete table of those summands from the point of view of (g,b)-…
Planar decomposition simplifies HOMFLY polynomial calculation for certain knots and links.
problem Calculating HOMFLY polynomial for specific types of knots and links.
method Planar decomposition of bipartite diagrams, lifting from sl(2) to sl(N).
result HOMFLY polynomials of many knots and links have planar decompositions.
The paper proves a theorem about shared Dehn surgeries between knots.
problem Understanding shared Dehn surgeries between distinct knots.
method Uses the JSJ decomposition of knot exteriors to provide an effective bound.
result Provides an effective bound on the maximal number of shared surgeries between knots.
This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.
Diagrammatic method calculates knot invariant from tangle decompositions.
problem Computing Rasmussen's invariant for knots.
method Diagrammatic approach using tangles and cobordisms.
result Computed s-invariants of all 3-strand pretzel knots. In this note, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, ξ) as the minimal genus of a page of an open book of M supporting the contact structu…
New method finds knots without low treewidth diagrams.
problem Finding knots without low treewidth diagrams.
method Structural graph theory and knot theory.
result Optimal obstruction for high representativity knots.
Torus decomposition shows foliation detected slopes for glued knot manifolds.
problem Detecting foliation detected slopes in glued knot manifolds.
method Torus decomposition and foliation analysis.
result Gluing knot manifolds identifies rational boundary slopes.
The paper shows knots can have zero support genus in certain spaces.
problem Understanding the support genus of Legendrian knots in specific spaces.
method Using open book decompositions and positive Dehn twists.
result Knots and links can have Legendrian representatives with support genus zero in certain spaces.
This paper decomposes generalized O'Hara's energies into components.
problem Decomposing generalized O'Hara's energies to understand their components.
method Using an analogue of Doyle-Schramm's cosine formula, the paper derives a decomposition for generalized O'Hara energies.
result Derives a decomposition for generalized O'Hara energies into three components.
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…
Strongly-cyclic branched coverings of knots are studied by using their (g,1)-decompositions. Necessary and sufficient conditions for the existence and uniqueness of such coverings are obtained. It is also shown that their fundamental groups admit geometric g-words cyclic presentations.
Study of 3-manifolds in 5-sphere using bridge decompositions.
problem Understanding embeddings of 3-manifolds in 5-sphere.
method Introduce and study bridge decompositions, use multisections of 5-manifolds.
result Every embedded 3-manifold admits a bridge decomposition.
A new method optimizes knot selection for spline dimensional decomposition in stochastic dynamic analysis.
problem Challenges in uncertainty quantification for dynamical systems with non-smooth or oscillating nonlinear behaviors.
method Interpolation-based optimal knot selection method for SDD, improving accuracy and computational efficiency.
result SDD with proposed knot selection yields higher accuracy than other methods, as shown in a lower control arm example.
Study of knot surgeries and JSJ decompositions to tackle L-space conjecture.
problem Understanding JSJ decompositions and L-space knots through Dehn surgeries. method Slope detection techniques applied to toroidal 3-manifolds and rational surgeries.
result JSJ graphs of certain knot exteriors are rooted intervals, implying left-orderable fundamental groups.
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.
New technique connects instanton Floer homology to Heegaard diagrams for knots and 3-manifolds.
problem Accessing structural properties of instanton Floer homology.
method Developed a new technique using Heegaard diagrams and sutures.
result Established a relation between instanton knot homology and Heegaard diagrams for (1,1)-knots. The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.
problem Existence of meridional essential surfaces in hyperbolic knot exteriors.
method Essential tangle decompositions and meridional essential embeddings.
result Infinite collection of hyperbolic knots with meridional essential surfaces of arbitrary genus and boundary components.
We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), when the substitute of the strand was a torus knot. We describe a general decomposition of satellite's colored HOMFLY in …
Study shows knots with coprime polynomials can't be concordant.
problem Primary decomposition of knot concordance.
method Using von Neumann rho-invariants and solvable filtration.
result Nonnegative integer n-solvable knots with coprime polynomials are not concordant.
A new version of ECK for rationally null-homologous knots, with a surgery formula.
problem Defining and computing ECK for rationally null-homologous knots.
method Generalized ECK to rational open book decompositions, proving independence of choices, and proving a surgery formula.
result Large negative n-surgery formula for ECK, supporting equivalence with knot Floer homology.
New method for knot group representations without polyhedral decompositions.
problem Representations of knot groups into PSL2(C). method Uses knot diagrams and a simple algorithm, avoiding triangulations.
result Explicit equations for canonical component of representations.
Every negative amphichiral knot is rationally slice.
problem Proving every negative amphichiral knot is rationally slice.
method Systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior.
result Every negative amphichiral link is rationally slice.
We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each factor). Underlying the theorem is an algebraic construction that also allows for the c…
In this paper, we define the rectangle condition on the bridge sphere for a n-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of 3-manifolds. We show that the satisfaction of the rectangle condition for a n-bridge decomposition can …
New formula for knot group representations and hyperbolic structures.
problem Understanding representations of knot groups and their geometric implications.
method Direct algebraic formula for geometric parameters of octahedral decompositions.
result Explicit criterion for critical points in Neumann-Zagier--Yokota potential function.