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.
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.
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.
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.
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.
The article calculates a multiplying factor to convert rational Vassiliev invariants to integer-valued ones.
problem Converting rational valued Vassiliev invariants to integer-valued ones.
method Calculates the minimal multiplying factor λ needed for rational Vassiliev invariants to become integer-valued.
result Obtains a set of integer-valued Vassiliev invariants.
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.
In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the M-th colored Jones polynomials evaluated at (N+1/2)-th root of unity with a fixed limiting ratio, s, of M and (N+1/2). We find out the…
We find approximations by Vassiliev invariants for the coefficients of the Jones polynomial and all specializations of the HOMFLY and Kauffman polynomials. Consequently, we obtain approximations of some other link invariants arising from the homology of branched covers of links.
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.
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 Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…
We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of link…
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.
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.
Study on the growth of colored Jones polynomial for figure-eight knot cables.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot cables.
method Analyzing the asymptotic growth of the N-dimensional colored Jones polynomial of a cable of the figure-eight knot. result The growth rate of the colored Jones polynomial is exponential and related to the Chern-Simons invariant.
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.
New methods reveal colored Jones polynomials from quantum R-matrices and knot invariants.
problem Understanding colored Jones polynomials of knots.
method Two realizations: quantum R-matrices and refined quantum modularity conjecture.
result New insights into knot invariants from quantum R-matrices and matrix conjectures.
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.
In this work, we give a formula for the logarithmic invariant of knots in terms of certain derivatives of the colored Jones invariant. This invariant is related to the logarithmic conformal field theory, and was defined by using the centers in the radical of the restricted quantum group at root of unity. A relation bet…
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.
The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural numb…
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…
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.
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.
Study uses big data to analyze quantum invariants.
problem Investigate structural properties of Jones polynomial.
method Exploratory and topological data analysis, including coloring, rank increase, categorification.
result Contrasts behavior of Jones polynomial under various enhancements.
Study knot invariants to deduce Hopf invariant and propose a slope conjecture.
problem Understanding the topological significance of knot invariants and their relations.
method Analyzing the Gukov-Manolescu knot series and its coefficients, relating to Hopf invariant and colored Jones polynomials.
result Explicit formula for the Hopf invariant in terms of colored Jones polynomials for fibered knots up to 12 crossings.
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's G2 quantum invariants. Our method consists in the study of Mark…
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.
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of `the categorification of the Jones polynomial'. For the same low cost we also provide some computations, including one that shows that Khovanov's…
New link homologies categorify Jones polynomial at odd prime powers.
problem Categorify Jones polynomial at odd prime powers.
method Specialize Cautis differential to positive integers.
result Non-isomorphic link homologies for odd primes.
In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial, the Kuperberg bracket, and the normalised arrow polynomial.
We generalize the colored Jones polynomial to 4-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.
We show examples of knots with the same polynomial invariants and hyperbolic volumes, with variously coinciding 2-cable polynomials and colored Jones polynomials, which are not mutants.
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.
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.
We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There i…
We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjectu…
We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo 4π2.
The study characterizes maximal subgroups of a Thompson group and bounds the change in a knot invariant.
problem Characterizing maximal subgroups of a Thompson group and understanding the change in a knot invariant.
method Characterization of maximal subgroups and analysis of knot invariant.
result The \( \vec{F} \)-index of a knot increases at most by 3 after orientation change.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Saddle point method applied to colored Jones polynomial.
result Asymptotic expansion formula for twist knots at given root of unity.
Paper categorifies Vassiliev skein relation for Khovanov homology.
problem Clarifying the relation between Vassiliev invariants and Khovanov homology.
method Developed a categorified version of Vassiliev skein relation on Khovanov homology.
result Khovanov homology's genus-one operation leads to a crossing change, enabling invariance under Reidemeister moves and extending to singular links.
Paper proves knots satisfy a conjecture using Jones polynomial.
problem Proving infinite families of knots satisfy the Cosmetic Surgery Conjecture.
method Computed Jones polynomial and invariants for two knot families.
result Two infinite families of knots satisfy the Purely Cosmetic Surgery Conjecture.
New theory for unoriented virtual links, extending classical invariants.
problem Invariants for unoriented virtual links not previously defined.
method Developed a new Khovanov homology theory for unoriented virtual links.
result Unoriented Jones polynomial for virtual links is a new invariant.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Asymptotic expansion formula for colored Jones polynomial using twist knots.
result Obtained asymptotic expansion formulas for twist knots at specified roots of unity.
We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distin…
Study on colored Jones polynomial of figure-eight knot for complex parameters.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot.
method Analyzing the asymptotic growth rate of the polynomial for complex parameters with small imaginary part.
result Growth rate of polynomial is related to the Chern-Simons invariant for large real part of the parameter and to the reciprocal of Alexander polynomial for small real part.
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.