Extends positive and almost positive links to successively almost positive ones.
problem Extending properties of positive and almost positive diagrams and links.
method Introducing successively almost positive diagrams and links, and analyzing their properties.
result Improves known results of positive and almost positive 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.
Characterizes a subset of links using quasipositive and homogeneous properties.
problem Understanding the properties of T-positive links.
method Characterization through strongly quasipositive and T-homogeneous braids.
result T-positive links are precisely the strongly quasipositive links that are closures of T-homogeneous braids.
We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's s-invariant. We also study almost positive links, in particular, determine the s-invariants of almost positive links. This result suggests that all almost positive links might be strongl…
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.
Study on quantum sl3 invariant for positive links.
problem Characterizing and understanding the quantum sl3 invariant of positive links. method Skein theory of sl3-webs, explicit formulae, diagrammatic quantities, obstructions. result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
Study of new link types and their invariants, extending previous results.
problem Properties of polynomial invariants and signatures of weakly successively almost positive links.
method Analysis of minimal genus and fibering properties, extending known theorems.
result Extension of Scharlemann-Thompson's theorem to weakly successively almost positive links.
The paper explores fibered and quasi-positive links, introducing new families and invariants.
problem Understanding the structure of L-space links and their properties. method Using the H-function as a concordance link invariant.
result Introduced a subfamily of fibered strongly quasi-positive L-space links and an infinite family of non-quasi-positive L-space links. Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
New bound on Jones polynomial for specific positive links.
problem Finding bounds on the Jones polynomial for positive links.
method Using previous results on positive fibered links, we found a new bound for a specific family of positive links.
result We provided a bound on the maximum degree of the Jones polynomial for positive links with a specific coefficient.
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.
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.
New bounds on Jones polynomial positivity for specific links.
problem Determining when Jones polynomial of positive links is non-negative.
method Developed new bounds and used them to obstruct positivity for infinitely many almost-positive diagrams.
result Infinitely many knots are classified as almost-positive.
Positive Thompson links are arborescent tangles.
problem Understanding the structure of Thompson group elements.
method Analyzing closures of bipartite arborescent tangles.
result Positive Thompson links correspond to arborescent tangles.
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.
In this paper, we show that a link which has a positive and almost alternating diagram is alternating, besides that a positive and non-alternating Montesinos link has an almost positive-alternating diagram.
New bounds on Khovanov homology for positive links families.
problem Bounding the maximum non-vanishing quantum degree of Khovanov homology.
method Diagram-independent upper bounds constructed using Khovanov homology.
result Constructs upper bounds on Khovanov homology, providing positivity obstructions.
Perturbing non-minimal bridge positions of a knot ensures similar behavior for its cable links.
problem Ensuring similar non-minimal bridge positions for cable links.
method Perturbing non-minimal bridge positions of a knot K and analyzing (2,2q)-cable links L. result Every non-minimal bridge position of a (2,2q)-cable link L is perturbed. 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.
Proves alternating quasipositive links are positive and strongly quasipositive.
problem Characterizing alternating quasipositive links.
method Analyzing alternating diagrams with specific crossing conditions.
result Alternating quasipositive links are positive and strongly quasipositive.
It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if the…
The study establishes a link between fibered links and their concordance invariants.
problem Determining the conditions for fibered strongly quasi-positive links.
method Proved that an n-component fibered link L is strongly quasi-positive if and only if τ(L)=g3(L)+n−1. result Explicitly determined fibered prime links with at most 9 crossings and their properties.
Study Khovanov homology of positive links and L-space knots, finding vanishing conditions.
problem Understanding Khovanov homology of positive links and L-space knots.
method Analyzing Khovanov homology groups in specific gradings and extending results to (p,q)-cables and Heegaard Floer L-space knots.
result Khovanov homology of positive links and L-space knots vanishes under certain conditions.
New links identified with Alexander polynomial signs to detect satellite links.
problem Detecting satellite links using Alexander polynomial coefficients.
method Introduced a set of links including positive braids and arborescent Hopf plumbings. Showed links in this set have opposite signs for leading and second coefficients of Alexander polynomials.
result Certain satellite links, like (n,1)-cables, are not in the set of links with opposite sign coefficients.
Researchers found infinite links with specific bridge positions.
problem Finding links with minimal bridge positions.
method Applying Takao et al.'s criterion to create links with locally minimal n-bridge and globally minimal m-bridge positions. result Provided an infinite family of links with specific bridge positions.
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.
Paper explores link and plat presentations, showing equivalence under bridge isotopy.
problem Relationship between links in bridge position and plat presentations.
method Established equivalence between Hilden double coset classes of plat presentations and bridge positions up to isotopy.
result Revealed equivalence of Hilden double coset classes for n-bridge unknot and torus knots in plat position.
New upper bound on Jones polynomial for fibered positive links.
problem Classifying positive and non-positive knots of crossing number ≤ 12.
method Proved a new upper bound on the maximum degree of Jones polynomial for fibered positive knots.
result Maximum degree of Jones polynomial for fibered positive knots is at most four times the minimum degree.
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.
An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus and 3-genus of a positive knot are equal. In this paper, we p…
Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.
problem Understanding when the Morton-Franks-Williams inequality holds for positive knots and links.
method Combinatorial characterisation and generating examples.
result Examples of diagrams achieving crossing number, braid index, and maximal self-linking number.
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.
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…
Standard position for surfaces extended to weakly generalized alternating links.
problem Proving standard position for surfaces in a broader class of links.
method Extending techniques from classical alternating links to weakly generalized alternating links.
result All weakly generalized alternating links are prime, and Conway spheres interact with diagrams as in classical setting.
We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.
The Links-Gould polynomial of alternating knots is shown to be log-concave and positive.
problem Verifying the positivity and log-concavity of the Links-Gould polynomial for alternating knots.
method Formulated a conjecture and verified it computationally for all 51.3 million knots with up to 19 crossings.
result All but 544 knots satisfy a stronger log-concavity condition.
We give a method for searching for thin positions of a given link.
We associate an open book with any connected plane checkerboard graph, thus providing a common extension of the classes of prime positive braid links and positive tree-like Hopf plumbings. As an application, we prove that the link type of a prime positive braid closure is determined by the linking graph associated with…
Study finds symplectic fillings' properties for specific contact covers.
problem Determining symplectic fillings' Euler characteristics and signatures.
method Analyzing exact symplectic fillings of contact branched covers.
result Identified links' symplectic fillings' properties.
Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case it has positive Conway polynomial (that is, all its coefficients…
Study links' arc index and Turaev genus, proving conjectures.
problem Understanding the arc index and Turaev genus of links.
method Computed arc index, established bounds, and conjectured inequalities.
result Proved conjectures linking crossing number, arc index, and Turaev genus.
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so - in fact they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously construc…
The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling, therefore its Thurston-Bennequin number is maximal among Legendrian …
In this paper, we prove that a link which has an almost positive diagram with a certain condition is Lagrangian fillable.
We define a family of link concordance invariants {sn}n=2,3,⋯. These link concordance invariants give lower bounds on the slice genus of a link L. We compute the slice genus of positive links. Moreover, these invariants give lower bounds on the link splitting number of a link. Especially, t…
We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere (M,ξ), whenever ξ is tight. More specifically, we show that the self-linking number of a transverse link T in (M,ξ), such that the boundary of its tubular neighbourhood …
Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…