Jones constructs knots from Thompson group elements.
problem No specific problem stated; focuses on constructions.
method Constructions of knots from Thompson group elements.
result Introduced two types of knots: oriented and unoriented.
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48 results for “Vaughan Jones”
Classifies isotopy classes of links from Thompson's group F and its subgroup.
problem Classify isotopy classes of links from Thompson's group F and its subgroup.
method Introduced a method to produce links from elements of Thompson's group F and its subgroup.
result Classified isotopy classes of links from Thompson's group F and its subgroup.
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.
The faithfulness of the orthogonal group case of Brauer's representation of the Brauer centralizer algebras restricted to their Temperley-Lieb subalgebras, which was established by Vaughan Jones, is here proved in a new, elementary and self-contained, manner.
In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group . In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an orient…
New method counts link components from Thompson group elements.
problem Counting connected components of positive Thompson links.
method Defining Thompson permutations from Thompson group elements.
result Number of orbits equals number of link components.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations
Jones rational coincidencesmath.GT
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.
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 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.
We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.
problem Categorify Jones-Wenzl projectors for odd Khovanov homology.
method Develop grading multicategories to replace grading categories, proving existence and uniqueness of categorified projectors.
result Existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology.
The work of Jonesmath.GT
Jones Polynomial shows unity in math.
problem No specific problem stated.
method Discussion of Jones Polynomial.
result Illustrates unity between different mathematical fields.
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…
Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between -polynomial and colored Jones polynomial.
method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between -polynomial and colored Jones polynomial potential function.
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 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.
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…
We prove the Farrell-Jones conjecture for free-by-cyclic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture.
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.
The Jones unknot conjecture states that the Jones polynomial distinguishes the unknot from nontrivial knots. We prove it for knots up to 23 crossings.
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 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 , 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…
Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …
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 shows colored Jones invariants limit to link volumes.
problem Volume conjecture for colored Jones invariants.
method Deformation of hyperbolic structure for link complements.
result Limits of colored Jones invariants related to link volumes.
New bounds on Jones polynomial positivity for specific links.
problem Determining when Jones polynomial of positive links is non-negative.
method Developed new bounds and used them to obstruct positivity for infinitely many almost-positive diagrams.
result Infinitely many knots are classified as almost-positive.
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…
It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alt…
A state generating is introduced to determine the Jones polynomial of a link. Formulae for two infinite families of knots are shown by applying this method, the second family of which are proved to be non-alternating. Moreover, the method is generalized to compute the Jones-Kauffman polynomial of a virtual link. As exa…
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.
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 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 links, focusing on .
result Jones polynomial is trivial if and only if the knot is trivial.
We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds , where or , which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…
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 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.