Satellite links of fully positive braids are characterized.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
A method to convert pretzel links into braids.
problem Representing pretzel links as braids.
method General procedure to convert any pretzel link into a braid.
result A specific braid word corresponding to the pretzel link.
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
Study improves HOMFLY polynomial coefficients for positive braid links.
problem Determining HOMFLY polynomial coefficients for positive braid links.
method Using geometric invariants like maximum Euler characteristics, number of split and prime factors.
result Improvements in known results for Conway and Jones polynomials of positive braid links.
Extended welded links are a generalization of Fenn, Rimányi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kam…
Positive braids have endless filling possibilities.
problem Understanding exact Lagrangian fillings of positive braid links.
method Analyzing positive braid Legendrian links and their fillings.
result Positive braid Legendrian links have infinitely many exact Lagrangian fillings.
Satellite links with many twists have simpler companions.
problem Relationship between satellite and companion links' complexity.
method Constructing satellite links with multiple full twists and analyzing their companion links.
result Satellite links with many twists have simpler companion links.
Positive braid knots have simple knot Floer homology.
problem Computing knot Floer homology for positive braids.
method Computed the next-to-top term of knot Floer homology.
result Rank is 1 for any prime positive braid knot.
Paper extends link Floer homology detection to almost braided links.
problem Detecting links that are nearly braided using link Floer homology.
method Inspired by nearly fibered knots, uses clasp-braids and topological characterisation.
result Khovanov homology detects Whitehead link and infinite families of links.
Study negative band numbers in braids and links.
problem Characterizing braids with negative band numbers.
method Using Birman, Ko, and Lee's left-canonical form.
result Characterize strongly and almost strongly quasipositive braids.
The paper finds linked periodic orbits in disc homeomorphisms using braids.
problem Finding linked periodic orbits in disc homeomorphisms.
method Interpreting linking of orbits by induced braids and using forcing relations.
result New examples of linked orbits of periods at most 4 for pseudo-Anosov braid types.
Study cobordism distances between 3-braid links and trefoil knots.
problem Understanding the geometric relationship between 3-braid links and trefoil knots.
method Determined cobordism distances between 3-braid links and trefoil knots, and explored limits of Coxeter's braid group result.
result Found cobordism distances between 3-braid links and trefoil knots, up to a constant error.
Single-colored ADO-3 invariant matches Links-Gould polynomial for 5-braid closures.
problem Matching ADO-3 invariant with Links-Gould polynomial for specific knot types.
method Proved for closures of 5-braids, conjectured for all knots and links.
result Single-colored ADO-3 invariant equals Links-Gould polynomial for 5-braid closures.
It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a lin…
We show that every quasipositive link has a quasipositive minimal braid representative, partially resolving a question posed by Orevkov. These quasipositive minimal braids are used to show that the maximal self-linking number of a quasipositive link is bounded below by the negative of the minimal braid index, with equa…
New method shows links can be braided open book bindings.
problem Understanding fibered links and their bindings.
method Mutual arc presentations and braided open books.
result Every fibered link is the binding of a braided open book.
Homogeneous quasipositive links are positive if their Seifert circles match braid index.
problem When are homogeneous quasipositive links positive?
method Showed that a specific type of diagram is positive.
result Homogeneous diagrams with matching Seifert circles and braid index are positive.
The paper classifies links up to link-homotopy using claspers.
problem Classifying links up to link-homotopy.
method Using Habiro's clasper calculus, defining a linear representation of the homotopy braid group, and providing a geometric proof.
result Geometric proof of Levine's classification of 4-component links and further classification of 5-component links in the algebraically split case.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…
This paper determines the braid indices for non-alternating pretzel links.
problem Determining the braid index of non-alternating pretzel links.
method Classified pretzel links into three types and computed braid indices for Type 1 and Type 2.
result Precise braid indices for all non-alternating Type 1 and Type 2 pretzel links.
The study establishes conditions for positive and quasi-positive links.
problem Characterizing and testing positive and quasi-positive links.
method Proves necessary conditions for link concordance and positivity.
result Characterizes positive links with unlinking number 1 and 2, and tests positive links as closures of positive braids.
New link detection results using closures of 3-braids.
problem Link detection using homology theories.
method Closure operations on 3-braids and homology theories.
result Detection of specific links using link Floer homology, Khovanov homology, and annular Khovanov homology.
Formula for spectrum linking braid and bridge indices.
problem Understanding the relationship between braid and bridge indices.
method Foliation theory applied to booklink embeddings.
result Formula for link spectra of braid and bridge indices.
We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid wo…
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of J-reflection groups. result Link groups of torus necklaces are precisely braid groups of J-reflection groups, with meridians as braid reflections. The paper extends knot theory to twisted virtual braids and links.
problem Generalizing knot theory to include twists.
method Introduced twisted virtual braids and proved theorems for twisted links.
result The Alexander and Markov theorems were extended to twisted links.
Study of generalized knots and links, proving inequality involving crossing number and braid index.
problem Proving an inequality involving the minimal crossing number and braid index for generalized knots and links.
method Introducing generalized crossings and moves, proving inequality for generalized knots and links.
result Proved inequality involving total crossing number and braid index for generalized knots and links.
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic table…
Study completes braid index determination for all pretzel links.
problem Determine braid indices of all pretzel links.
method Divided pretzel links into three types and derived formulas for braid indices.
result Close to determining braid indices for all Type 3 pretzel links.
The paper defines plat closures for spherical braids and shows links in RP3 can be realized this way.
problem Defining and analyzing plat closures for spherical braids in RP3. method Defining plat closures, associating residual permutations, and presenting moves on spherical braids.
result The number of components of the plat closure link of a spherical braid is equal to the number of disjoint cycles in its residual permutation.
New 'book links' generalize braids and plats, proving Markov's theorem.
problem Generalizing braid theory to new structures.
method Extending open book foliations to prove Markov's theorem for new objects.
result Proved Markov's theorem in a broader context of 'book links'.
We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.
Using the band representation of the 3-strand braid group, it is shown that the genus of 3-braid links can be read off their skein polynomial. Some applications are given, in particular a simple proof of Morton's conjectured inequality and a condition to decide that some polynomials, like the one of 9_{49}, are not adm…
Adding a braid closure to a fibered knot makes a link ribbon concordance minimal.
problem Finding the minimal ribbon concordance for links in S3. method Link Floer homology combined with classical techniques.
result Any link L in S3 can be made ribbon concordance minimal by adding a single unknot. The paper explores connections between braids, links, and cobordisms using algebraic methods.
problem Investigating functions on manifolds and their connections to braids, links, and cobordisms.
method Algebraic methods including group theory, sheaves, and formal groups.
result Constructs Lazard's one-dimensional universal commutative formal group and applies it to cobordism theory.
The closure of a braid in a closed orientable surface Σ is a link in Σ×S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close t…
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.
In this paper we describe braid equivalence for knots and links in a 3-manifold M obtained by rational surgery along a framed link in S3. We first prove a sharpened version of the Reidemeister theorem for links in M. We then give geometric formulations of the braid equivalence via mixed braids in S3 using the…
We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T2-knots and turned spun T2-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…
Study of Legendrian links using Floer theory and cluster varieties.
problem Understanding exact Lagrangian fillings of positive braid Legendrian links.
method Floer-theoretic approach and exact Lagrangian cobordisms.
result Proves that positive braid Legendrian links admit infinitely many exact Lagrangian fillings.
For an oriented surface link S, we can take a satellite construction called a 2-dimensional braid over S, which is a surface link in the form of a covering over S. We demonstrate that 2-dimensional braids over surface links are useful for showing the distinctness of surface links. We investigate non-trivial examp…
New theorem allows transverse links to be braided with rational book structure.
problem Transverse isotopy of links in contact 3-manifolds.
method Generalization of Pavelescu's argument for rational open book decomposition.
result Every transverse link can be isotoped to a braid with rational open book.
Paper shows pseudometrics on braid groups are nondegenerate.
problem Stability of braid types defined by symplectic morphisms.
method Uses Hofer metric to define pseudometrics and shows stability under small distance.
result Pseudometrics are nondegenerate.
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads and length of this braid does not exceed n(4n-5)+2.
The paper defines subgroups of camomile type and studies singular braids and links.
problem Understanding subgroups of camomile type in singular braid groups.
method Presentations, defining relations, and group theory.
result The center of SGn is a direct factor in SPn but not in SPn. The notion of free link is a generalized notion of virtual link. In the present paper we define the group of free braids, prove the Alexander theorem that all free links can be obtained as closures of free braids and prove a Markov theorem, which gives necessary and sufficient conditions for two free braids to have the…
Invariants defined for braid systems under Hurwitz equivalence.
problem Invariants for braid systems under Hurwitz equivalence.
method Crossing matrix and polynomial invariant introduced.
result Invariant defined for surface braids and surface links.
It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.