Study Vassiliev invariants for virtual knots, expanding quantum theory.
problem Understanding Vassiliev invariants for virtual knots.
method Define chord diagrams, weight systems, and Lie algebra weight systems for rotational virtual knots.
result Extended quantum invariants capture more information than standard invariants.
Squeezed knots are slices of minimal cobordisms; obstructions come from quantum knot invariants.
problem Characterizing and obstructing squeezed knots.
method Analysis of cobordisms, quantum knot invariants, and stable cohomology operations.
result Effective obstructions to squeezedness come from quantum knot invariants, notably Rasmussen invariant refinements.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
problem Constructing stronger quantum knot invariants.
method Developing quantum cocycle invariants using Yang-Baxter cohomology and deformation theory.
result Quantum cocycle invariants yield stronger invariants in certain examples.
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor produ…
Topological quantum computers use hyperbolic knots for computations.
problem The difficulty of calculating quantum invariants of knots.
method Using hyperbolic knots to compute topological quantum computer invariants.
result The hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.
Paper connects knot invariants and Morse flow loops.
problem Connecting quantum group invariants and Morse flow loops for knot study.
method Defining a two-variable series invariant by counting Morse flow loops in knot complements and proving it agrees with quantum group BPS series.
result Correspondence proven for all braid-homogeneous knots.
We propose a gauge model of quantum electrodynamics (QED) and its nonabelian generalization from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants from quantum field theory based on the Chern-Simon Lagrangian. From our approach we ca…
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, stu…
This review connects knot invariants to quiver representations.
problem Relating knot invariants to quiver representations.
method Relates symmetric quivers and their partition functions to quantum invariants of knots.
result Establishes a correspondence between knot invariants and quiver representations.
We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.
Study on quantum invariants of twist knots using saddle point method.
problem Asymptotic expansion of Reshetikhin-Turaev invariants of twist knots.
method Saddle point method applied to integral q-surgery. result Asymptotic expansion formula for Reshetikhin-Turaev invariants.
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting R-matrix of Uq(sl2) as cluster transformation, introducing auxiliary parameter ε. result Derives perturbed-Alexander invariants with higher-order terms in ε. The fundamental problem of knot theory is to know whether two knots are equivalent or not. As a tool to prove that two knots are different, mathematicians have developed various invariants. Knots invariants are just functions that can be computed from the knot and depend only on the topology of the knot. Here we descri…
Quantum model for knotted graphs from knot theory.
problem Constructing an isotopy invariant polynomial for knotted bipartite ribbon graphs.
method Applying quantum topology to construct an isotopy invariant polynomial.
result Computed the expected number of loops in the double dimer model.
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…
Authors prove quantum invariant conjecture for figure-eight knot complement.
problem Connecting quantum invariants of surface diffeomorphisms to hyperbolic volumes.
method Analyzes the simplest case of a one-puncture torus and figure-eight knot complement.
result Proves conjecture linking quantum invariant to hyperbolic volume.
Quantum invariants of 3-manifolds and links reviewed, with connections to other topological invariants.
problem Quantum invariants of 3-manifolds and links.
method Review of recent developments and connections to other invariants.
result Rich features of quantum invariants like quantum modularity and Verma module structures.
In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes c…
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.
The abstract discusses resurgent functions in quantum knot invariants.
problem Understanding the asymptotic expansion of quantum knot invariants.
method Using resurgent functions and q-series to conjecture and compute knot invariants. result Explicit computations match conjectured values for specific knots.
Study on quantum invariants from surgeries on torus knots.
problem Quantum invariants of three-manifolds from surgeries along torus knots.
method Analysis of Witten-Reshetikhin-Turaev invariant and Chern-Simons invariants.
result Quantum invariants can be described as sums of Chern-Simons invariants and twisted Reidemeister torsions.
New series invariant for knots and cables, with robustness and relations.
problem Computing series invariants for complex knots and cables.
method Explicit computation and analysis of satellite knots, including a cable of the figure eight knot.
result First example of a cable knot with more than ten crossings, demonstrating robustness and integrality.
In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a m…
New quantum invariants for planar knotoids improve knot classification.
problem Classifying and distinguishing planar knotoids with up to five crossings.
method Define biframed planar knotoids and construct new invariants.
result Improved classification of planar knotoids with up to five crossings.
Quantum trace map defines invariants for knots and links, confirming a length conjecture.
problem Defining invariants for knots and links in hyperbolic 3-manifolds.
method Introducing a quantum trace map for ideally triangulated knot complements, combining with state-integral models.
result Perturbative invariants determine an asymptotic expansion of the Jones polynomial, confirming the length conjecture.
New approach to quantum knot invariants using perturbed Gaussian generating functions.
problem Developing universal quantum knot invariants.
method Introducing generating functions of the form PeG where G is quadratic and P is a perturbation, and developing a calculus for such functions. result The rank one invariant ZD dominates sl2-colored Jones polynomials and relates to knot genus and Whitehead doubling. 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.
Study calculates quantum hyperbolic invariants for figure-eight knot complement, finding it either 0 or half the volume.
problem Computing quantum hyperbolic invariants for knot complements.
method Computed the real part of the semi-classical limit of quantum hyperbolic invariants of the figure-eight knot complement.
result The real part is rigid and either 0 or half the hyperbolic volume of the knot complement.
We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also …
This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework.
Extends biquandle brackets to psyquandles for knot and pseudoknot invariants.
problem Counting invariants for singular and pseudoknots.
method Define quantum enhancements of psyquandle counting invariant.
result Proper quantum enhancements for singular and pseudoknots.
New proof and formula linking fusion trees to quantum knot invariants.
problem Quantum knot invariants encoding in non-semisimple TQC.
method Connection between fusion trees and Lawrence representations, using graphical calculus.
result Explicit encoding of quantum knot invariants via fusion trees.
Verifies a conjecture for the figure eight knot.
problem Relates A-ideal and recurrence ideal of knots.
method Uses quantum A-ideals, q-holonomicity, and AJ conjecture.
result Strong AJ conjecture verified for figure eight knot.
In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tam…
Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
problem Integrality structure of framed knots' quantum invariants.
method Explicit formulas of colored HOMFLY-PT invariants of torus knots, verified in a limit form for any framed knots.
result Proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
Modified knotoids with framing and coframing for quantum invariants.
problem Defining and classifying knotoids with framing.
method Defining framed and biframed knotoids, showing topological correspondence, and constructing quantum invariants.
result Generalized quantum knotoid invariants constructed.
We introduce \textit{Kaestner brackets}, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle 2-cocycle invariants…
This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective …
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.
Study shows most knots up to 10 crossings can't be chirally cosmetic.
problem Identifying knots that can undergo chirally cosmetic surgeries.
method Used invariants from 3-manifold theory, including quantum SO(3)-invariant and Heegaard Floer homology.
result Approximately 75% of knots up to 10 crossings do not admit chirally cosmetic surgeries.
Unified invariant of knots derived from Verma modules.
problem Constructing a unified invariant of knots from quantum sl2.
method Braid groups' action on tensors of Verma modules.
result Unified invariant interpolates colored Jones and ADO polynomials.
The paper sets genus bounds for twisted quantum invariants.
problem Bounding the degree of twisted quantum invariants for knots.
method Using Reshetikhin-Turaev construction and Drinfeld doubles.
result Degree of polynomials is bounded by 2g(K)⋅d(H). Automates machine learning of correlations between knot invariants.
problem Discovering and validating new relationships between knot invariants.
method Trained a neural network on 200,000 sets of knot invariants to predict an output invariant.
result Found novel correlations not explained by known results in knot theory.
This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is formulated in terms of the biquandle, a generalization of the quandle of David Jo…
We introduce a Poincaré polynomial with two-variable t and x for knots, derived from Khovanov homology, where the specialization (t,x) = (1,−1) is a Vassiliev invariant of order n. Since for every n, there exist non-trivial knots with the same value of the Vassiliev invariant of order n as that of the…
We provide a geometric construction of the boundary states for handlebodies which we in turn use to give a geometric formula for the Witten-Reshetikhin-Turaev quantum invariants. We then analyze the asymptotics of this invariant in the special case of a three manifold given by 1-surgery on a knot and we show that if th…
The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.
problem Quantum invariants of knots with additional structure.
method Using twisted Drinfeld doubles and Reshetikhin-Turaev invariants.
result Reidemeister torsion of knot complements as a quantum invariant.