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.
New lower bound for knot genus using Links-Gould invariant.
problem Finding a tighter lower bound for knot genus.
method Representation theory of Uqgl(2∣1) to prove degree of Links-Gould polynomial bounds Seifert genus. result The Links-Gould polynomial provides a new lower bound on knot genus, detecting specific knots like Kinoshita-Terasaka and Conway.
This paper gives a connection between well chosen reductions of the Links-Gould invariants of oriented links and powers of the Alexander-Conway polynomial. We prove these formulas by showing the representations of the braid groups we derive the specialized Links-Gould polynomials from can be seen as exterior powers of …
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.
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.
Formula for colored Links-Gould polynomial with genus bounds.
problem Calculating polynomial for knots colored with specific representations.
method Cabling formula and genus bounds for the Links-Gould polynomial.
result Genus bounds and specialization to Alexander polynomial for colored Links-Gould polynomial.
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.
Proves a plumbing-multiplicative property of a Links-Gould invariant.
problem Proving a multiplicative property of the Links-Gould invariant.
method Using Laurent polynomials and topological plumbing of surfaces.
result The Links-Gould invariant is monic in the Laurent polynomial ring.
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.
The Links--Gould polynomial detects causality in spacetimes.
problem Detecting causality in spacetimes using link invariants.
method Using the Links--Gould polynomial, which specializes to the Alexander--Conway polynomial.
result The Links--Gould polynomial distinguishes Allen-Swenberg links and detects causality.
Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to Uqsl(2) at fourth roots of unity, or by considering the super Hopf algebra Uqgl(1∣1). In this paper, we show …
The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant of oriented (1,1) tangles, which is derived from the representation of the braid generator associated with the one-parameter family of four dimensional representations with highest weights (0,0|a) of the quantum superalgebra U_q[gl(2|1)]. We u…
We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (de…
The exterior algebra of a vector space admits a family of braided Hopf structures.
problem Identifying the exterior algebra with a Nichols algebra and studying its braided Hopf structures.
method Explicit computation of structure constants and construction of solutions to the Yang-Baxter equation.
result The exterior algebra of a vector space admits a one-parameter family of braided Hopf structures.
We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.
This paper describes a method for the automatic evaluation of the Links-Gould two-variable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at most 5. Data are present…
In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link …
New knot polynomials reveal patterns and mutations.
problem Understanding the structure and behavior of knot polynomials.
method Using a specific multivariable polynomial invariant derived from Nichols algebras.
result Emerging patterns in knot polynomials, including genus bounds and unexpected mutations.
We show that the Alexander-Conway polynomial Delta is obtainable via a particular one-variable reduction of each two-variable Links-Gould invariant LG^{m,1}, where m is a positive integer. Thus there exist infinitely many two-variable generalisations of Delta. This result is not obvious since in the reduction, the repr…
It is known that the first two-variable Links--Gould quantum link invariant LG≡LG2,1 is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations o…
New knot polynomials derived from Nichols algebras and braided Hopf algebras.
problem Developing new knot invariants from algebraic structures.
method Constructing knot invariants from solutions to the Yang--Baxter equation over generalized Yetter--Drinfel'd modules.
result Reproduces known knot polynomials and discovers new multivariable invariants.
This paper describes a method to obtain state model parameters for an infinite series of Links-Gould link invariants LG^{m,n}, based on quantum R matrices associated with the (\dot{0}_m | \dotα_n) representations of the quantum superalgebras U_q[gl(m|n)]. Explicit details of the state models for the cases n=1 and m=1,2…
In this paper we study the tensor powers of the standard representation of the quantum super-algebra Uq(sl(2∣1), focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by LGn. Their dimensions were conjectured by I. Marin and E. Wagner \cite{MW}. We prove this conjecture, desc…
Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are …
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.
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.
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.
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. 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…
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.
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomial…
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.
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.
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…
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…
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.