New invariant for tied links connects states without resolution dependence.
problem Understanding the Kauffman-like states for tied links.
method Defined Aicardi-Juyumaya states and showed their contribution to the invariant is independent of resolution.
result The double bracket of a tied link diagram can be computed and used to find linked but differently polynomial tied links.
Paper derives explicit formulas for AJ-bracket of tied links.
problem Lack of state-sum formula for AJ-bracket of tied links.
method Analyzed AJ-states of 2- and 3-tied link diagrams, derived resolution trees, and state-sum formulas.
result Derives first closed-form expressions for AJ-bracket.
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…
Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.
problem Asymptotic behavior of colored Jones polynomials of figure-eight knot.
method Analyzes growth rate of polynomial evaluated at specific points.
result Growth rate determined by Chern-Simons invariant of an affine representation.
We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial a…
In this paper we introduce the tied links, i.e. ordinary links provided with some ties between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpret…
New monoids tied to symmetric group and Jones/Brauer monoids discovered.
problem Understanding monoids attached to symmetric group and related monoids.
method Introduced ramified monoids and tied-like monoids, provided presentations.
result Found new monoids that cannot be described as ramified monoids.
We introduce a two-parameters bt-algebra which, by specialization, becomes the one-parameter bt-algebra, introduced by the authors, as well as another one-parameter presentation of it; the invariant for links and tied links, associated to this two-parameter algebra via Jones recipe, contains as specializations the inva…
We introduce the concept of tied links in the solid torus, which generalize naturally the concept of tied links in S3 previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and subsequently we recover this invariant by using Jones' method over the bt-…
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.
New method proves Jones Polynomial's connect sum property.
problem Jones Polynomial's behavior under connect sums.
method Trip matrix method for calculating Jones Polynomial.
result Jones Polynomial is multiplicative under connect sums.
Researchers compute and predict knot volumes using colored Jones polynomials.
problem Computing and predicting volumes of hyperbolic knots.
method Vertex model approach, neural network training, polynomial evaluations.
result 3-colored Jones polynomials predict knot volumes with high accuracy.
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. This paper will be an exposition of the Kauffman bracket polynomial model of the Jones polynomial, tangle methods for computing the Jones polynomial, and the use of these methods to produce non-trivial links that cannot be detected by the Jones polynomial.
Upper bound on Jones polynomials density modulo primes.
problem Density of Jones polynomials modulo prime numbers.
method Derived an upper bound on Jones polynomials density within a large degree range.
result Upper bound on Jones polynomials density modulo primes.
Survey on categorifying Jones polynomial.
problem Categorification of Jones polynomial.
method Not explicitly stated, likely involves algebraic and geometric categorification techniques.
result Significance and ramifications in geometry, algebra, and topology.
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.
In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras Yd,n(q), which are not topologically equivalent to the Homflypt polynomial. We then present the algebra FTLd,n(q) which is the appropriate Temperley-Lieb analogu…
New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume …
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.
New methods assess topological entanglement in periodic systems.
problem Assessing topological entanglement in systems with periodic boundary conditions.
method Introducing Periodic Jones polynomial and Cell Jones polynomial.
result Periodic Jones polynomial is a recurring factor of Jones polynomial of finite cutoffs.
Jones polynomials have infinitely many roots of unity as zeros.
problem Finding roots of unity as zeros of Jones polynomials.
method Constructing families of prime knots with specific Jones polynomials.
result Infinitely many roots of unity are zeros of some Jones polynomials.
The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the Volume and certain sums of coeff…
A new knot invariant uses permutations to extend Jones polynomials.
problem Extending Jones polynomials to classical and virtual knots and links.
method Colorings by permutations of a finite set to define new knot invariants.
result Established properties and computed polynomials for small cases.
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…
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.
Jones polynomial coincidences explored for rational knots.
problem Identifying coincidences in Jones polynomial of rational knots.
method Moves on continued fraction expansion of rational knots, conjectured to generate all coincidences.
result Conjectured moves are sufficient to generate all Jones rational coincidences.
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative re…
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.
Paper explores the Jones polynomial and its impact on knot theory and related fields.
problem Exploring the Jones polynomial and its applications in knot theory.
method Recalling the Jones polynomial and its development, discussing its connections to various mathematical and physical contexts.
result The Jones polynomial has wide-ranging applications and connections in mathematics and physics.
Novel Jones polynomial for open curves in 3D space.
problem Measuring entanglement complexity of open curves in 3-space.
method Defining Jones polynomial for linkoids and extending to collections of open and closed curves.
result Jones polynomial for open curves has real coefficients and is continuous.
Categorifies Jones polynomial using Lie theory.
problem Categorifying the coloured Jones polynomial.
method Lie theoretic categorification with Jones-Wenzl projectors.
result Constructs a categorification of the coloured Jones polynomial.
Paper extends Cohen's method to compute Jones polynomial for certain braid subfamilies.
problem Computing Jones polynomial for specific knot families.
method Using weighted adjacency matrices and determinants for certain subfamilies of braid groups.
result Jones polynomial can be computed in polynomial time for certain subfamilies of braid groups.
Jones slopes detect figure eight knot, and characterize alternating knots.
problem Detecting knots using Jones polynomials.
method Strong slope conjecture and colored Jones polynomials.
result Jones slopes detect figure eight knot and characterize alternating knots.
Globalizes Jones and Alexander polynomials using topological intersections.
problem Link invariants from graded intersections of Lagrangians.
method Topological model proving the Jones polynomial's well-definedness and constructing globalizations.
result Proves the Jones polynomial and constructs globalizations of Jones and Alexander polynomials.
Jones Polynomial shows unity in math.
problem No specific problem stated.
method Discussion of Jones Polynomial.
result Illustrates unity between different mathematical fields.
Paper defines new versions of Jones polynomial and Khovanov homology.
problem No specific problem stated; focuses on new definitions.
method Using maps from Gauss diagrams to their variants to define new Jones polynomial and Khovanov homology.
result New versions of Jones polynomial and Khovanov homology behave differently from original ones.
Study shows quantum modularity in figure-eight knot's colored Jones polynomial.
problem Asymptotic behavior of colored Jones polynomial of figure-eight knot.
method Analyzing polynomial evaluated at specific points and showing asymptotic equivalence.
result Quantum modularity demonstrated in the figure-eight knot's colored Jones polynomial.
The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing defin…
This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false thet…
In previous joint work with Frohman and Lofaro a noncommutative generalization of the A-polynomial of a knot was introduced, consisting of a finitely generated ideal of polynomials (the noncommutative A-ideal) in the quantum plane. The present paper shows that the noncommutative A-ideal of a knot, together with finitel…
This article contains general formulas for Tutte and Jones polynomials for families of knots and links given in Conway notation and "portraits of families"-- plots of zeroes of their corresponding Jones polynomials.
New knot models analyze local entanglement for robust curve analysis.
problem Lack of local structural information in classical knot theory.
method Proposed multiscale and persistent Jones polynomials.
result Models are stable to small perturbations, robust for real-world applications.
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.
Using the Huynh and Le quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that for a knot which is the closure of a positive braid, the first N coefficients of…
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.
Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.
problem Computing Jones polynomial for a specific family of links.
method Computed Jones polynomial for T((p,q),(2,s)) links, focusing on s=2n. result Jones polynomial is trivial if and only if the knot is trivial.