Paper computes Alexander polynomials for arborescent links.
problem Explicit formulas for Alexander polynomials are hard to compute for most link families.
method Efficient method for arborescent links, using recursive polynomials.
result Explicit closed formulas for pretzel links derived.
The paper studies polynomials and ideals from colored Jones polynomials for links.
problem Understanding the structure of colored Jones polynomials for links.
method Investigates commutative and noncommutative ideals derived from colored Jones polynomials.
result Formulates the link version of the AJ conjecture.
New skein theory for Links-Gould polynomial simplifies link evaluations.
problem Computing Links-Gould polynomial for oriented links.
method Developed a cubic braid-type skein theory.
result Skein theory can evaluate any oriented link.
New polynomial invariant distinguishes singular links.
problem Distinguishing singular links using existing invariants.
method Generalized quandle polynomial to singquandles and constructed a singular link invariant.
result New polynomial invariant distinguishes singular links with same counting invariant.
New link polynomials linked to cluster theory.
problem Connecting link polynomials to cluster theory.
method Introducing new link polynomials and their expansion over perfect matchings.
result Bracket polynomials of certain links can be realized as specializations of cluster variables.
Generalized quandle polynomial used for stuquandles, stuck links, and RNA folding.
problem Defining polynomial invariants for stuquandles, stuck links, and RNA foldings.
method Introduced a generalized quandle polynomial and proved its invariance for stuquandles. Used this invariant to define polynomials for stuck links and RNA foldings.
result Polynomial invariants for stuquandles, stuck links, and RNA foldings.
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.
We construct a 2-variable link polynomial, called WL, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine WL…
New proof limits Jones polynomial values for quasi-alternating links.
problem Limits on Jones polynomial values for quasi-alternating links.
method Proved finitely many values of Jones polynomial for quasi-alternating links of a given determinant.
result Only finitely many quasi-alternating links have a given Jones polynomial.
The paper improves bounds on the complexity of computing link polynomials.
problem Computing link polynomials by the skein relation is complex.
method Proved new upper and lower bounds on skein tree depth.
result New bounds on skein tree depth are stronger than previous ones.
Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.
problem Understanding polynomial invariants of pretzel links.
method Revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links P(1,1,n). result Reveals properties of Alexander-Conway and Kauffman bracket polynomials for P(1,1,n). Formula calculates MOY webs and link polynomials.
problem No specific problem stated; focuses on evaluation.
method Closed formula for exterior webs and link polynomials.
result Closed formula for evaluating MOY webs and link polynomials.
New polynomial invariants for virtual links are stronger than F-polynomials.
problem Defining new invariants for virtual links.
method Introducing weight functions and a recurrent construction for new invariants.
result New polynomial invariants are stronger than F-polynomials.
Extended Thistlethwaite's result on Jones polynomials of quasi-alternating links.
problem Characterizing Jones polynomials of quasi-alternating links.
method Analyzing structure and properties of Jones polynomials for quasi-alternating links.
result Jones polynomials of prime quasi-alternating links have no gaps.
This paper investigates the equivalence between Yamada polynomial and Jones polynomial of associated links for brunnian θ-curves.
problem Understanding the relationship between Yamada polynomial and Jones polynomial for θ-curves.
method Investigates the equivalence between the normalized Yamada polynomial of θ-curves and the Jones polynomial of their associated links.
result Shows that the two polynomials are equivalent for brunnian θ-curves.
The paper defines new polynomials for links and linkoids.
problem No specific problem stated; focus on new polynomials.
method Defined as sums over states of link or linkoid diagrams with f=n. result Constructed new polynomials for starred links and linkoids.
Study Alexander polynomials of links in 3-torus.
problem Investigate Alexander polynomials of links in 3-torus.
method Diagrammatic approach, Reidemeister moves, fundamental group, homology group, Alexander polynomials, twisted Alexander polynomials.
result Computed Alexander and twisted Alexander polynomials of links in 3-torus.
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.
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 Alexander polynomials of special alternating links and generalize Fox's conjecture.
problem Distinguish special alternating links up to isotopy using polynomial invariants.
method Combinatorial and discrete geometric properties of Alexander polynomials of special alternating links.
result Generalized Alexander polynomials of special alternating links can be expressed in terms of volumes of root polytopes of unimodular matrices.
Empirical evidence suggests link polynomials can detect causality in spacetimes.
problem Detect causality in (2+1)-dimensional globally hyperbolic spacetimes. method Introduced a new invariant of certain tangles related to the Conway polynomial.
result The Conway polynomial does not detect causality in certain spacetime scenarios.
Study links and quivers, proving polynomial equality conjecture.
problem Link and quiver invariants and their relations.
method Cluster algebra invariants, point count polynomials, skein relations.
result Equality conjecture between plabic graph link polynomial and quiver point count polynomial proved for specific cases.
Alexander quandles fail to distinguish certain links, thus not detecting causality.
problem Detecting causality in spacetimes using link polynomials.
method Examined Alexander quandles' ability to distinguish specific links.
result Alexander quandles cannot distinguish the connected sum of two Hopf links and Allen-Swenberg Links.
New formulas derived for Jones polynomial of rational links.
problem Calculating the Jones polynomial of rational links.
method Colored Brylawski's tensor product formula for Tutte polynomials, finite automaton for crossing signs.
result Generalization of existing formulas for rational links.
The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links is valid for virtual links which admit checkerboard colorings.
Study of knots and links in 2-complexes, defining linking numbers and polynomials.
problem Understanding knots and links in 2-dimensional complexes.
method Definition of linking numbers and Kauffman-type bracket polynomials for links in 2-complexes.
result Established relationships between 2-complexes and knots/links in 3-manifolds.
The paper extends knot polynomials to annular and toroidal pseudo links.
problem Analyzing pseudo links with undefined crossings.
method Introducing Kauffman bracket and Jones-type polynomials for annular and toroidal pseudo links.
result New tools for studying annular and toroidal pseudo links.
Link signature limit depends on linking matrix under specific polynomial condition.
problem Limits of Tristam-Levine signature function under precise polynomial conditions.
method Analysis of Alexander polynomial and linking matrix.
result Limit of Tristam-Levine signature at 1 determined by linking matrix under specific polynomial condition.
In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author. The relation between this…
New invariant for virtual links defined using homology.
problem Invariants for virtual links and their properties.
method Defining homological arrow polynomial and using it to study virtual links.
result New invariant for virtual links and its properties.
Paper introduces a new skein relation for multivariable polynomials of virtual links.
problem Understanding properties of virtual links through polynomial invariants.
method Developed a virtual skein relation for multivariable polynomials of virtual links.
result New skein relation for multivariable polynomials of virtual links.
A new method calculates HOMFLY-PT polynomials for bipartite links.
problem Computing HOMFLY-PT polynomials for bipartite links efficiently.
method Generalizes Goeritz matrix method for bipartite links.
result Reduces HOMFLY-PT polynomial calculation to matrix algebra.
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented s…
All link types arise from semiholomorphic polynomials.
problem Proving every link type can be represented by semiholomorphic polynomials.
method Constructive proof showing every link type arises from a weakly isolated singularity of a semiholomorphic polynomial.
result Every link type in the 3-sphere arises as the link of a weakly isolated singularity of a semiholomorphic polynomial.
Study on colored Jones polynomial and link complements.
problem Understanding the structure of link complements with arbitrary colors.
method Investigated the potential function of the colored Jones polynomial and established a relationship with hyperbolicity.
result Evidence supports the Chen-Yang conjecture on link complements.
Character variety of Borromean link solved, Alexander polynomial formula found.
problem Character variety of Borromean link
method Determined irreducible SL(2,C) character variety, found Alexander polynomial formula
result Formula for twisted Alexander polynomial on character variety
The Links-Gould invariant of alternating links has log-concave coefficients.
problem Log-concavity of Links-Gould coefficients for alternating links.
method Experimental and computational evidence.
result The Links-Gould coefficients of alternating links are log-concave.
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect chirality and even non-invertibility of virtual knots and links. Furthermore, it …
Paper conjectures Links-Gould invariant generalizes Alexander polynomial.
problem Classifying knots and links using the Links-Gould invariant.
method Analyzing classical properties of the Links-Gould invariant.
result Evidence suggests Links-Gould invariant provides lower bounds for genus and fiberedness criteria.
We prove that twisting any quasi-alternating link L with no gaps in its Jones polynomial VL(t) at the crossing where it is quasi-alternating produces a link L∗ with no gaps in its Jones polynomial VL∗(t). This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other th…
Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…
Dye and Kauffman defined surface bracket polynomials for virtual links by use of surface states, and found a relationship between the surface states and the minimal genus of a surface in which a virtual link diagram is realized. They and Miyazawa independently defined a multivariable polynomial invariant of virtual lin…
New bounds on HOMFLY polynomial for homogeneous links.
problem Bounding the minimum v-degree of HOMFLY polynomial for homogeneous links. method Proved a slice version of Cromwell's inequality and a related conjecture.
result New bounds on the minimum v-degree of HOMFLY polynomial for homogeneous links. We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these polynomials for knots, and for links with any given number of components.
We give an algorithm for computing the Teichmüller polynomial for a certain class of fibered alternating links associated to trees. Furthermore, we exhibit a mutant pair of such links distinguished by the Teichmüller polynomial.
We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperb…
We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials…
Topological model created for HOMFLY-PT polynomial from link diagrams.
problem Constructing categorifications for HOMFLY-PT polynomial.
method Explicit Lagrangian submanifolds on Heegaard surfaces.
result Invariants derived from link diagrams are given by intersections of submanifolds.