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…
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.
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…
This paper calculates the growth constant for quantum knot mosaics.
problem Calculating the growth constant for quantum knot mosaics.
method Introduced knot mosaic system, defined quantum knots, and used enumeration to find the growth constant.
result Existence and bounds of the knot mosaic constant δ. Researchers developed an algorithm to count all graph mosaics.
problem Defining and counting graph mosaics to represent graph diagrams.
method Using a recursion formula of state matrices and sixteen graph mosaic tiles.
result Produced the exact enumeration of all graph mosaics.
This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere …
Quantum knots and knotted zeros linked through complex plane mappings.
problem Understanding knotted zeros in quantum states of hydrogen.
method Classifying maps from 3-space to complex plane, relating to quantum knots and lattice structures.
result Every smooth knot in 3-space has a corresponding smooth map to the complex plane with a knotted inverse image of zero.
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.
Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m×n matrix of mosaic tiles which are T0 through T10 depicted as below, representing a knot or a link b…
Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m×n matrix of mosaic tiles (T0 through T10 depicted in the introduction) re…
Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n×n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K i…
New types of knot mosaics help count and analyze knots efficiently.
problem Counting and analyzing knots efficiently.
method Introducing period and toroidal knot mosaics and developing algorithms for their enumeration.
result Exact enumeration of period knot mosaics and asymptotics of toroidal knot mosaics.
Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n×n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining pr…
Paper introduces equations to distinguish knots without using knot invariants.
problem Distinguishing knots using only topological and combinatorial methods.
method Linear systems of equations derived from HOMFLYPT and Kauffman polynomials.
result First examples of distinguishing knots without knot invariants.
We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" …
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.
Knot theory applied to quantum computing models.
problem Using knot theory for quantum computing models.
method Exploring knot theory applications in quantum computing.
result Knot theory introduces topological concepts to quantum computing.
New central elements found in a quantum algebra related to knot theory.
problem Understanding the algebraic structure of SLd-skein algebras. method Threaded polynomials from symmetric functions.
result Extraction of central elements in SLd-skein algebra. 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.
Fundamental weight systems identified as quantum states.
problem Identifying which weight systems are quantum states.
method Analyzing the Cayley distance kernel on the symmetric group and its positivity.
result All fundamental gl(n)-weight systems are quantum states.
Survey of quantum enhancements in knot theory.
problem Classifying knots using quantum invariants.
method Collecting and analyzing various quantum invariants.
result New invariants defined by coloring knots with algebraic objects.
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 invariants help distinguish knots via algebra and equations.
problem Determine if two knots are equivalent.
method Use Hopf algebras and solutions to the Yang-Baxter equation.
result Quantum invariants are powerful tools for knot classification.
Develops method to construct Lie algebra weight system kernel using Vogel algebra.
problem Detecting correlators and distinguishing knots in 3D Chern-Simons theory.
method Uses Vogel's Λ algebra and Jacobi diagrams.
result Explicitly provides Jacobi diagrams in the kernel of sl_N weight system.
A new mosaic system for immersed surface-links is introduced.
problem Constructing a mosaic system for immersed surface-links.
method Using singular marked graph diagrams, the mosaic number for immersed surface-links is defined and discussed.
result A mosaic system for immersed surface-links is established.
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.
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.
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…
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.
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 Aq polynomial. 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.
A new quantum invariant for virtual knots and links.
problem Quantum invariants for virtual knots and links.
method Generalization of biquandle brackets to parity biquandles.
result The new invariant is stronger than classical biquandle brackets for virtual 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…
Quantum method categorifies knot polynomial.
problem Alexander polynomial of knots.
method Modified foam evaluation, algebraic approach.
result Quantum categorification of Alexander polynomial.
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 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.
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.
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.
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.
Study of singular knots connects knot theory with quantum algebra.
problem Understanding the structure of knots with transverse double points.
method Analyzes singular knots and their relationship to Vassiliev invariants and quantum algebra.
result Extensions of non-numerical knot invariants to singular knots have been explored.
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.
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.
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.
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 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.
A new quantum relation connects exceptional Lie algebras and knots.
problem Understanding the relationship between exceptional Lie algebras and quantum invariants of knots.
method Developed a two-parameter skein relation on trivalent graphs that specializes to exceptional Lie algebras.
result Found a new quantum exceptional polynomial that agrees with classical computations for knots and links.
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.
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 …