New invariant of virtual knots generalizes existing polynomial invariants.
problem No specific problem stated; focuses on a new invariant.
method Described a new transcendental function invariant of virtual knots.
result Generalizes several polynomial invariants of virtual knots.
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.
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…
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.
New link invariants from colored link polynomial.
problem Constructing stronger link invariants.
method Using skein invariant of colored links.
result Invariants stronger than HOMFLYPT polynomial.
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.
We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants which further generalize the quandle counting invariant.
New algebraic setup defines quantum link invariants.
problem Defining and controlling quantum link invariants.
method Quantum Schur--Weyl duality and variants.
result Global definitions of quantum polynomials.
New knotoid invariants defined for classical and virtual knotoids.
problem Defining invariants for knotoids and virtual knotoids.
method Constructing new invariants including odd writhe, parity bracket polynomial, affine index polynomial, and arrow polynomial.
result Affine index polynomial and arrow polynomial provide bounds on height of classical knotoids.
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.
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.
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. Review of invariants for spatial graphs.
problem No specific problem stated; review of existing invariants.
method Combinatorial and polynomial invariants of spatial graphs.
result Overview of Alexander polynomial, fundamental quandle, and Yamada polynomial.
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.
Defines a new polynomial invariant for virtual links.
problem Developing a new invariant for virtual links.
method Introduces a multi-variable Affine Index Polynomial.
result Proves the invariant is a Vassiliev invariant of order one.
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.
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…
New invariant defined for colored links via skein relation.
problem Defining a polynomial invariant for colored links.
method Defining a polynomial invariant via a skein relation.
result Specializes to the Jones polynomial for classical links.
Formulae for Vassiliev invariants derived from Kauffman polynomial.
problem Computing Vassiliev invariants from knot polynomials.
method State model of Kauffman polynomial, Gauss diagram identities, arrow diagram identities.
result Gauss diagram formulae for Vassiliev invariants of order 3.
Polynomial invariant derived from birack labelling of knots.
problem Developing a polynomial invariant for a broader class of knot theories.
method Generalizing biquandle colouring to birack labelling, reducing to biquandle invariant.
result Polynomial invariant for a class of knot theories.
Two new polynomial invariants for long virtual knots.
problem Defining new polynomial invariants for long virtual knots.
method Introducing V1(K;t) and V2(K;t), establishing properties, and showing realizability. result First derivatives of V1(K;t) and V2(K;t) at t=1 define finite type invariants of degree three. Paper defines a polynomial invariant for surface-links using quantum A_2 invariant.
problem Defining a polynomial invariant for surface-links.
method Using the quantum A_2 invariant and Yoshikawa moves, a polynomial is defined for marked graph diagrams.
result The polynomial invariant is useful for studying ribbon 2-knots.
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 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.
New polynomial invariants for links identified from Yokonuma-Hecke algebras.
problem Defining new polynomial invariants for classical links.
method Using a Markov trace on Yokonuma-Hecke algebra of type A.
result New invariants distinguish knots from links and are stronger than Homflypt polynomial.
Enhances knot and link invariants using quandle modules.
problem Distinguishing knots and links using polynomial invariants.
method Integrates quandle modules into the quandle coloring quiver.
result The enhanced invariant distinguishes knots and links.
New link invariants derived from quandle filtrations.
problem Constructing link invariants from quandle structures.
method Decategorification of subquandle filtrations of quandle coloring quivers.
result Definition of disrespectful and graded disrespectful polynomials.
We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants which generalize the quandle counting invariant.
The paper connects ADO polynomials to Vassiliev invariants for knots.
problem Connecting ADO polynomials to Vassiliev invariants for knots.
method Exploiting the colored Jones polynomials and their decomposition as Vassiliev invariants, the authors transpose this to ADO polynomials.
result A unique computable expansion of ADO polynomials as Vassiliev invariants.
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 polynomial invariants for knots and links from quandle coloring quiver decategorification.
problem Defining new polynomial invariants for knots and links.
method Decategorification of the quandle coloring quiver to create polynomial invariants.
result The invariants are not determined by the quandle counting invariant.
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.
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.
Proves partial-dual genus polynomial is a knot invariant weight system.
problem Proving the partial-dual genus polynomial is a weight system.
method Proved the polynomial satisfies the four-term relation, thus making it a weight system.
result The partial-dual genus polynomial is a Vassiliev knot invariant weight system.
New knot invariants created using multiple skein relations.
problem Creating more powerful knot invariants.
method Introducing new ways to smooth crossings and using a system of skein equations.
result A simplified version of the modified invariant is easier to compute and generalizes the two-variable Kauffman polynomial.
Study connects group invariants through outer automorphisms and polynomial relations.
problem Understanding polynomial invariants of free-by-cyclic groups.
method Introducing orientable fully irreducible outer automorphisms to relate McMullen polynomial and Alexander polynomial.
result Characterization of when homological stretch factor equals geometric stretch factor.
Alexander invariant created for doodles, vanishes on unlinked doodles.
problem Creating an Alexander type invariant for doodles.
method Deformation of Tits representation and Chebyshev polynomials of second kind.
result Invariant vanishes on unlinked doodles with more than one component.
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…
Algorithm finds knot and link invariants using q,p-numbers.
problem Finding invariants of knots and links.
method Algorithm based on q,p-numbers in bosonic two-parameter quantum algebra.
result Derives generalized Alexander polynomials and Jones polynomials.
Introduce a two-variable parity polynomial for virtual knotoids
problem Define a polynomial invariant for virtual knotoids
method Based on the parity of classical crossings
result Can distinguish pairs not distinguished by odd writhe and affine index polynomial
New polynomials defined for quandle structures, enhancing graph invariants.
problem Enhancing the counting invariant for spatial graphs and handlebody-links.
method Introducing quandle polynomials and G-family polynomials for quandles, defining enhancements for invariants.
result New enhancements of the G-family counting invariant for trivalent spatial graphs and handlebody-links.
Paper connects Links-Gould invariants to Alexander polynomial powers.
problem Understanding relationships between Links-Gould invariants and Alexander polynomial powers.
method Derived specialized Links-Gould polynomials from braid group representations, showing they are exterior powers of Burau representations.
result Proved formulas connecting Links-Gould invariants to powers of the Alexander-Conway polynomial.
A new polynomial invariant for strongly involutive links.
problem Characterizing strongly involutive links using polynomial invariants.
method Introducing a two-variable polynomial invariant \(P^e\) with equivariant skein relations.
result Specialisation of \(P^e\) recovers the graded Euler characteristic of a spectral sequence.
Expands Jones polynomial for Legendrian knots with categorification.
problem Polynomial invariants for Legendrian knots.
method Introduces new skein relation and categorifies polynomial invariant.
result Natural extension of Jones polynomial and Khovanov homology for Legendrian knots.
The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynom…
New knots share same Upsilon invariant despite different Alexander polynomials.
problem Identifying concordant knots via Upsilon invariant.
method Examined hyperbolic L-space knots and their Upsilon invariants.
result Infinitely many pairs of hyperbolic L-space knots with distinct Alexander polynomials share the same Upsilon invariant.
The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek poly…
A n-dimensional Lie group G equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on G. Relatively to this affine structure we show that the left invariant Poisson tensor π+ corresponding to $\om^+$ is po…