Study on unimodality of plucking polynomial with delay function.
problem Exploring unimodality of plucking polynomial with delay function.
method Presented a formula for the plucking polynomial of hedgehog rooted trees and explored unimodality with specific delay functions.
result Found interesting examples and speculations on unimodality of plucking polynomials with delay functions.
Classifies trees with strictly unimodal q-polynomials.
problem Classifying rooted trees with strictly unimodal q-polynomials.
method Classification based on plucking polynomials and criteria for trapezoidal shapes.
result Generalizes results on strict unimodality of q-binomial coefficients.
The paper defines when two trees have identical plucking polynomials.
problem Identifying trees with the same plucking polynomial.
method Provided a sufficient and necessary condition for trees with identical plucking polynomials.
result A sequence of non-negative integers can be realized as a rooted tree.
New method finds coefficients of Catalan states using plucking polynomials.
problem Finding coefficients of Catalan states in lattice crossings.
method Using plucking polynomials of rooted trees with delay functions.
result Coefficients of Catalan states are unimodal sequences.
New formulas derived for lattice crossing coefficients, improving computation efficiency.
problem Computing coefficients of Catalan states in lattice crossings.
method Using plucking polynomial and Θ_A-state expansion, deriving new properties and formulas.
result Coefficients of Catalan states factor under specific conditions, leading to more efficient computation.
Paper finds coefficients of Catalan states using Θ_A-state expansion.
problem Finding coefficients of Catalan states of lattice crossings.
method Uses Θ_A-state expansion to express coefficients as a linear combination of other states.
result Shows that coefficients can be found using Θ_A-state expansion.
Scattering transform improves note onset detection and instrument recognition in music transcription.
problem Note onset detection and instrument recognition in music transcription.
method Multiscale scattering operators applied to MIDI-driven datasets and real musical pieces.
result Scattering transform outperforms other sound representations for note onset detection and instrument recognition.
New F-polynomial distinguishes knotoid diagrams not previously possible.
problem Polynomial invariants for knotoids.
method Introduced F-polynomial and constructed distinguishing examples. result New invariant F distinguishes knotoids not previously possible. 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.
Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.
problem Examining relationships between Affine Index Polynomial and Sawollek Polynomial.
method New approach to extract Affine Index Polynomial from Sawollek Polynomial.
result Constructs a concise proof of Mellor's Theorem.
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
Characterizes skein polynomial for links and knots.
problem Understanding skein polynomial for links and knots.
method Avoids smoothing of crossings to characterize polynomials.
result Characterizations of polynomials for knots and links.
Extends interior polynomial to signed bipartite graphs and connects to HOMFLY polynomial.
problem Invariants of signed bipartite graphs and their relation to HOMFLY polynomial.
method Extending interior polynomial to signed bipartite graphs and showing equality to HOMFLY polynomial part.
result Interior polynomial of signed bipartite graphs equals part of HOMFLY polynomial for planar case.
Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.
problem Understanding and characterizing knot polynomials from Gaussian calculus.
method Gaussian calculus of generating series for noncommutative algebras, connected sum of knots.
result Half of the polynomials vanish and three polynomials are explicitly given.
The paper defines and classifies Cappell-Shaneson polynomials.
problem Characterizing Cappell-Shaneson polynomials.
method Algebraic conditions on polynomials, reduction modulo primes, and construction of infinite series.
result Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5, and several infinite series of degree 6.
Developed algorithms to compute three polynomial invariants of veering triangulations.
problem Computing polynomial invariants of veering triangulations.
method Introduced and used algorithms for taut, veering, and Teichmüller polynomials based on upper and lower tracks of veering triangulations.
result Proved that the lower and upper taut polynomials are equal but the veering polynomials can differ.
New interpretation of Alexander polynomial for virtual knots.
problem Determining writhe polynomial for virtual knots.
method New interpretation of Alexander polynomial and use of it to define a second-order writhe polynomial.
result Alexander polynomial determines writhe polynomial for virtual knots.
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.
Algorithm calculates polynomial coefficients of link invariants.
problem Computing first coefficients of link invariants.
method Dynamic programming algorithm for Homflypt and Kauffman polynomials.
result Polynomial time complexity for first coefficients.
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). Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between A-polynomial and colored Jones polynomial. method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between A-polynomial and colored Jones polynomial potential function. Generalizes index polynomial to virtual tangles.
problem No specific problem stated; generalization of polynomial invariant.
method Generalized index polynomial to virtual tangles.
result Three polynomial invariants result from the generalization.
Polynomial time knot invariant discovered.
problem Finding efficient knot invariants.
method Developed a polynomial time algorithm for a strong knot invariant.
result Strongest known knot invariant computed in polynomial time.
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…
We find the degree of the Hilbert polynomial for HOMFLYPT homology.
problem Understanding the growth and structure of HOMFLYPT homology.
method Analyzing the degree of the Hilbert polynomial for closed braids.
result The degree of the Hilbert polynomial is l−1 for a braid with l components. 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 taut polynomial equals a twisted Alexander polynomial.
problem Understanding the relationship between taut polynomials and Alexander polynomials.
method Defined taut polynomial of veering triangulations and proved it equals a twisted Alexander polynomial.
result The taut polynomial equals a twisted Alexander polynomial of the underlying manifold.
Quantum polynomials are derived from a specific tribracket structure.
problem Quantum enhancement polynomials for oriented links.
method Defined using a canonical two-element tribracket, proving polynomials can be derived from five specific ones.
result Universal quantum enhancement polynomials are strictly stronger than the Jones polynomial.
New polynomials show all knots of a certain type have simple structure.
problem Classifying knots with specific properties.
method Introduced new L-polynomials and F-polynomials for virtual knots, and defined totally flat-trivial knots.
result All Akimova-Matveev knots are totally flat-trivial and their affine index polynomials can be calculated.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
Innovates polynomial invariant for tribrackets.
problem Counting and distinguishing knots and links.
method Introduces subtribracket polynomials and uses them to enhance counting.
result Enhanced counting invariant for knots and links.
Researchers extend Alexander polynomial to knotoids and linkoids.
problem Defining and studying Alexander polynomial extensions for knotoids and linkoids.
method Developed and proved conjecture on mock Alexander polynomial for knotoids and linkoids.
result Proved conjecture on mock Alexander polynomial for knotoids and linkoids.
Unified ADO and colored Jones polynomials for knots.
problem Determining ADO polynomials from colored Jones polynomials.
method Constructing a two-variable knot invariant using completions of rings and algebra.
result Unified invariant maps colored Jones polynomials to ADO polynomials.
New invariants for tied links extend classical polynomial invariants.
problem Defining new invariants for tied links.
method Extending Kauffman and Jones polynomials for tied links.
result New invariants are more powerful than existing ones.
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 Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.
problem Determining Alexander polynomials for ribbon and virtual knots.
method Using ribbon's intrinsic singularity information, defining half Alexander polynomial, and developing simplified formulas.
result New formulas for Alexander polynomials of general knots and virtual knots in terms of Gauss diagrams.
Alexander polynomials relate to KP hierarchy via 1-hook property.
problem Relating knot polynomials to the KP hierarchy.
method Kontsevich construction and linear equations reformulation.
result Solutions of reformulated system induce KP equations in Hirota form.
New invariant dominates Jones, Kuperberg, and arrow polynomials.
problem Dominating known knot invariants.
method Picture formalism leading to a new invariant.
result New invariant outperforms Jones, Kuperberg, and arrow polynomials.
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…
We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that nsata-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, gen…
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.
This paper generalizes knot polynomials to include the HOMFLY polynomial.
problem Generalizing knot polynomials to include the HOMFLY polynomial.
method Using path posets to directly generalize the construction of the Jones and Alexander polynomials to the HOMFLY polynomial.
result The HOMFLY polynomial is obtained by specializing path posets.
Paper solves a central question about nonnegative polynomials related to isoparametric polynomials.
problem Whether a given nonnegative polynomial is a sum of squares of polynomials.
method Solves the problem completely for nonnegative polynomials associated with isoparametric polynomials.
result The paper provides a complete solution for the specific case of isoparametric polynomials.
Jones polynomials derived from K-theory of a cluster algebra.
problem Jones polynomials of knots and links.
method K-theory of a cluster C*-algebra of the sphere with two cusps.
result Interplay between Chebyshev and Jones polynomials.
We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial. Along the way we also calculate the Kauffman bracket skein module of all 2-brid…
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.
Study exact polynomials to compute Mahler measure and relate it to volume function.
problem Compute Mahler measure of exact polynomials.
method Define volume function on vanishing set, prove local extrema on 2D torus, derive Mahler measure formula.
result Prove Mahler measure of irreducible exact polynomials is greater than volume function amplitude.
Polynomials with similar modulus levels are bilipschitz equivalent.
problem Understanding when polynomials are bilipschitz equivalent.
method Examined polynomials with moduli up to bilipschitz equivalence.
result Level curves of polynomials are bilipschitz equivalent.