Quantum physics model uses knot theory for fragile topology.
problem Modeling quantum physics' fragile topology.
method Knot theoretic algorithm.
result Quantum physics' fragile topology modeled.
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48 results for “quantum knot theory”
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.
This paper explores the interactions between knot theory and quantum computing. On one side, knot theory has been used to create models of quantum computing, and on the other, it is a source of computational problems. Knot theory is often used to introduce topological idea to people without a formal mathematical backgr…
Quantum trace map connects Teichmüller theory and quantum groups.
problem Connecting quantum groups to Teichmüller theory for knots.
method Quantum snakes technology to relate Fock-Goncharov monodromy matrices to quantum SL_n.
result Quantized Fock-Goncharov matrices satisfy quantum SL_n relations.
A quantum N-dimer modelmath.QA
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.
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.
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as polynomial.
In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and suc…
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.
Simplified geometric derivation of quantum A-polynomials for knots.
problem Deriving quantum A-polynomials for knots in a simple geometric way.
method Geometric derivation using Ward identities in Chern-Simons theory, contact geometry, and Kauffman calculus.
result Simplified presentation of quantum A-polynomials, making them accessible to a broader audience.
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…
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…
Quantum Knots and Mosaicsquant-ph
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantu…
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.
Holomorphic quantum modular forms linked to knot volumes.
problem Understanding algebraic properties of quantum modular forms.
method Analyzing descendant state integrals for specific knots.
result Illustrated algebraic properties for the (-2,3,7)-pretzel knot.
Parabolic skein modulesmath.QA
Develops skein theory for 3-manifolds with defects, extending quantum character stacks.
problem Quantum character stacks and their applications in 3-manifolds with surface defects.
method Parabolic induction/restriction for quantum groups, quantum decorated character stacks, ideal triangulations, gluing equations.
result Knot invariants related to quantum -polynomial, concrete computation method.
The abstract discusses resurgent functions in quantum knot invariants.
problem Understanding the asymptotic expansion of quantum knot invariants.
method Using resurgent functions and -series to conjecture and compute knot invariants.
result Explicit computations match conjectured values for specific knots.
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.
Knot invariants and higher representation theory II: the categorification of quantum knot invariantsmath.GT
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…
Unified theories for colored sl(2) knot homology.
problem Equivalence of different models for colored sl(2) knot homology.
method Conceptualized properties into a Chebyshev system and proved its uniqueness.
result Equivalence of Khovanov and Cooper-Krushkal models for the unknot.
Knot mosaic tabulationmath.GT
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…
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…
Study of knot complements yields quantum modularity insights.
problem Understanding quantum invariants of knot complements.
method Large-N analysis of -series invariants, counts of holomorphic curves.
result Closed-form expressions for -deformed for -torus knots.
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 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.
Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on `Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an matrix whose entries are eleven mosaic tiles, represent…
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting -matrix of as cluster transformation, introducing auxiliary parameter .
result Derives perturbed-Alexander invariants with higher-order terms in .
Virtual Knot Theorymath.GT
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss…
The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation …
New central elements found in a quantum algebra related to knot theory.
problem Understanding the algebraic structure of -skein algebras.
method Threaded polynomials from symmetric functions.
result Extraction of central elements in -skein algebra.
The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into .
result Alexander polynomial as the singular locus of a coherent sheaf.
We study the invariant of knots in lens spaces defined from quantum Chern-Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the paper, we propose a B-model topological string theory description of torus knots in…
In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of fla…
Psyquandle Bracketsmath.GT
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.
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.
Multi-dimensional state-integrals of products of Faddeev's quantum dilogarithms arise frequently in Quantum Topology, quantum Teichmüller theory and complex Chern--Simons theory. Using the quasi-periodicity property of the quantum dilogarithm, we evaluate 1-dimensional state-integrals at rational points and express the…
New quantum invariant is asymptotically multiplicative under cyclic covers.
problem Quantum invariants are not multiplicative under finite covers.
method Introduced a perturbative power series invariant of cusped hyperbolic 3-manifolds.
result The power series is asymptotically multiplicative under cyclic covers.
In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our approach uses a new concept, Burnside groups of links, which establishes unexpected re…
Enumeration on graph mosaicsmath.GT
Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors deve…
A Cable Knot and BPS-Seriesmath.GT
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.
Fast algorithm for braid group Hecke representation, applied to knot invariants.
problem Computing topological invariants of knots efficiently.
method Representation-theoretic approach to braid group, leveraging quantum topology.
result Fast algorithm for Hecke representation of braid group, finding non-trivial braids.
We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge grou…
We compute Khovanov homology for tangles using TQFT.
problem Khovanov homology for tangles is not well studied or computed.
method Topological Quantum Field Theory (TQFT) construction.
result A comprehensive method for computing Khovanov homology of tangles.
Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physicsmath.RT
The thesis explores centralisers and Hecke algebras in representation theory with applications to knots and physics.
problem Understanding centralisers and Hecke algebras in representation theory.
method Review of classical and quantum Schur-Weyl duality, discussion of quantum groups and their centralisers, and application to knot theory and physics.
result New insights into centralisers and Hecke algebras, leading to solutions for the Yang-Baxter equation and link invariants.
Turaev transformed knot theory and 3-manifold invariants.
problem Understanding knots and links invariants of 3-manifolds.
method Classical topology techniques and quantum topology.
result Introduced new ideas and tools in knot theory and 3-manifold invariants.
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.
Super-A-polynomialmath.AG
We review a construction of a new class of algebraic curves, called super-A-polynomials, and their quantum generalizations. The super-A-polynomial is a two-parameter deformation of the A-polynomial known from knot theory or Chern-Simons theory with SL(2,C) gauge group. The two parameters of the super-A-polynomial encod…
We study the Chern-Simons topological quantum field theory with an inhomogeneous gauge group, a non-semi-simple group obtained from a semi-simple one by taking its semi-direct product with its Lie algebra. We find that the standard knot observables (i.e. traces of holonomies along knots) essentially vanish, but yet, th…