Quantum invariants are explained as intersections in configuration spaces.
problem Quantum invariants of knots and links.
method Topological intersections in configuration spaces.
result Coloured Jones and Alexander polynomials are special cases of intersection pairings.
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.
New geometric invariant from disc intersections captures all coloured Jones polynomials.
problem Constructing a universal knot invariant from configuration spaces.
method Defining a new local system and Lagrangian submanifolds in the disc.
result The new invariant recovers Habiro's universal invariant and more.
Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.
In this paper we will present a homological model for Coloured Jones Polynomials. For each colour N∈N, we will describe the invariant JN(L,q) as a graded intersection pairing of certain homology classes in a covering of the configuration space on the punctured disk. This construction is based on the …
The paper connects quantum invariants to intersections of Lagrangians in symmetric power spaces.
problem Computing colored Jones and Alexander polynomials.
method Using two Lagrangians in a symmetric power of a surface to compute polynomials.
result Colored Jones and Alexander polynomials are special cases of a graded intersection between Lagrangians.
Unified quantum invariants via intersections of embedded Lagrangians.
problem Unified quantum invariants for Uq(sl(2)). method State sum of Lagrangian intersections in configuration spaces.
result Recovery of coloured Jones and Alexander polynomials.
New quantum knot invariants derived from Verma modules.
problem Constructing universal quantum knot invariants from Verma modules.
method Defining level N universal invariants from finite quotients of Verma modules over quotient rings.
result Maximal universal invariants for prime N, interpolating Jones and ADO 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.
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a O(N23cwpoly(n)) time algorithm to compute any Resh…
Characters from logarithmic VOAs linked to torus link invariants.
problem Understanding characters of logarithmic vertex operator algebras.
method Relating characters to coloured Jones invariants of torus links.
result Characters of logarithmic VOAs are limits of coloured Jones invariants of torus links.
In this paper we look for closed expressions to calculate the number of colourings of prime knots for given linear Alexander quandles. For this purpose the colouring matrices are simplified to a triangular form, when possible. The operations used to perform this triangularization preserve the property that the entries …
This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let π_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longit…
Motivated by a possible connection between the SU(N) instanton knot Floer homology of Kronheimer and Mrowka and sl(N) Khovanov-Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colourings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yam…
R-coloured knot polynomials for m-strand torus knots Torus[m,n] are described by the Rosso-Jones formula, which is an example of evolution in n with Lyapunov exponents, labelled by Young diagrams from R⊗m. This means that they satisfy a finite-difference equation (recursion) of finite degree. For…
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.
We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating …
Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group Uq(sl(2)) at roots of unity. This sequence recovers the original Alexander polynomial as the first term. We give a topological model for this invariants, showing that they ca…
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.
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.
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…