Researchers develop a new quantum invariant using a matrix dilogarithm for 3-manifolds.
problem Developing quantum invariants for 3-manifolds.
method Using a s l 3 \mathfrak{sl}_3 sl 3 matrix dilogarithm and quantum groups. result The s l 3 \mathfrak{sl}_3 sl 3 matrix dilogarithm can be considered as a 6 j j j -symbol. A new invariant for links generalizes Alexander polynomial for sl_3.
problem Defining a non-abelian generalization of the Alexander polynomial.
method Using quantum s l 3 \mathfrak{sl}_3 sl 3 representations and Laurent polynomials. result Established a direct relation between Δ s l 3 Δ_{\mathfrak{sl}_3} Δ sl 3 and the Alexander polynomial. Study on quantum s l 3 \mathfrak{sl}_3 sl 3 invariant for positive links.
problem Characterizing and understanding the quantum s l 3 \mathfrak{sl}_3 sl 3 invariant of positive links. method Skein theory of s l 3 \mathfrak{sl}_3 sl 3 -webs, explicit formulae, diagrammatic quantities, obstructions. result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
Quantum invariants from U h s l ( 2 ∣ 1 ) U_h\mathfrak{sl}(2|1) U h sl ( 2∣1 ) are q-holonomic.
problem Understanding quantum invariants from a specific quantum group.
method Demonstrated q-holonomic property through quantum group representations.
result Existence of an underlying field theory for these quantum invariants.
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
This article gives matrix factorizations for the trivalent diagrams and double line appearing in s l n \mathfrak{sl}_n sl n quantum link invariant. These matrix factorizations reconstruct Khovanov-Rozansky homology. And we show that the Euler characteristic of the matrix factorization for a double loop equals the quantum dimens…
Extends knot invariant computation to symmetrically colored sl_N.
problem Computing quantum knot invariants for s l N \mathfrak{sl}_N sl N . method Develops symmetrically colored R matrix for s l N \mathfrak{sl}_N sl N . result Defines F K s l N , s y m F^{\mathfrak{sl}_N, sym}_K F K sl N , sy m for positive braid knots. In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra s l ( 2 ∣ 1 ) \mathfrak{sl}(2|1) sl ( 2∣1 ) . This construction based on nilpotent irreducible finite dimensional representations of quantum group U ξ s l ( 2 ∣ 1 ) \mathcal{U}_ξ\mathfrak{sl}(2|1) U ξ sl ( 2∣1 ) where ξ ξ ξ is a root of unity of odd …
We show how to define invariants of graphs related to quantum s l ( 2 ) \mathfrak{sl}(2) sl ( 2 ) when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.
Quantum invariants derived from U q ( s l 2 ) \mathcal{U}_q(\mathfrak{sl}_2) U q ( sl 2 ) link holonomy.
problem Quantum invariants of links and their relations.
method Using quantum groups and Schur-Weyl duality, constructing quantum holonomy invariants.
result Quantum invariants of links can be derived from U q ( s l 2 ) \mathcal{U}_q(\mathfrak{sl}_2) U q ( sl 2 ) representations. We define and study the category of symmetric s l 2 \mathfrak{sl}_2 sl 2 -webs. This category is a combinatorial description of the category of all finite dimensional quantum s l 2 \mathfrak{sl}_2 sl 2 -modules. Explicitly, we show that (the additive closure of) the symmetric s l 2 \mathfrak{sl}_2 sl 2 -spider is (braided monoidally) equivalent to …
In this paper, we study the quantum s l ( n ) \mathfrak{sl}(n) sl ( n ) representation category using the web space. Specially, we extend s l ( n ) \mathfrak{sl}(n) sl ( n ) web space for n ≥ 4 n\ge 4 n ≥ 4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial P n ( q ) P_n(q) P n ( q ) specialized to a one variable polynomial …
Quantum traces map skein algebras to Fock-Goncharov spaces.
problem Establishing quantum traces between skein algebras and Fock-Goncharov spaces.
method Defining and proving properties of quantum traces for S L n SL_n S L n -skein algebras. result Existence and properties of quantum traces for S L n SL_n S L n -skein algebras. We develop a diagrammatic calculus for representations of unrolled quantum s l 2 \mathfrak{sl}_2 sl 2 at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given h…
Frobenius homomorphisms for SL_n skein modules generalize knot theory results.
problem Quantum group representations and skein theory for SL_n character varieties.
method Representation theory of quantum groups and skein theory.
result Frobenius homomorphisms for stated SL_n skein modules are defined and their properties are explored.
The abstract discusses new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
problem Developing new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
method Examining two m-traces in the category of representations over quantum s l ( m ∣ n ) \mathfrak{sl}(m|n) sl ( m ∣ n ) , considering quotients, and conjecturing generalizations. result Quotients of perturbative modules over quantum s l ( m ∣ n ) \mathfrak{sl}(m|n) sl ( m ∣ n ) lead to 3-manifold invariants and ETQFTs. Paper proves zero stability for one-row colored sl₃-Jones polynomials.
problem Stability of coefficients in colored Jones polynomials.
method Linear skein theory based on Kuperberg's sl₃-webs.
result Zero stability for B-adequate links with anti-parallel twist regions.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
New basis for quantum gl_N invariants derived from Macdonald polynomials.
problem Constructing new bases for quantum gl_N invariants.
method Using interpolation Macdonald polynomials and Okounkov's results.
result Cyclotomic expansions for gl_N invariants and knot invariants.
We study junctions of Wilson lines in refined SU(N) Chern-Simons theory and their local relations. We focus on junctions of Wilson lines in antisymmetric and symmetric powers of the fundamental representation and propose a set of local relations which realize one-parameter deformations of quantum groups $\dot{U}_{q}(\m…
Geometrically describes hyperbolic structures on link complements using quantum groups.
problem Describing hyperbolic structures on link complements algebraically.
method Uses octahedral decomposition and Kashaev-Reshetikhin's braiding on quantum group U ξ ( s l 2 ) \mathcal{U}_ξ(\mathfrak{sl}_2) U ξ ( sl 2 ) . result Shows how to interpret geometrically the algebraic gluing equations for hyperbolic structures.
We define a family of the braid group representations via the action of the R R R -matrix (of the quasitriangular extension) of the restricted quantum s l ( 2 ) \mathfrak{sl}(2) sl ( 2 ) on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the c…
Motivated by a possible connection between the S U ( N ) \mathrm{SU}(N) SU ( N ) instanton knot Floer homology of Kronheimer and Mrowka and s l ( N ) \mathfrak{sl}(N) sl ( N ) Khovanov-Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colourings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yam…
Extends quantum trace map to SL3(C) for 3D surfaces.
problem Generalizing quantum trace map to higher dimensions.
method Definition of SL3(C) quantum trace invariant.
result Construction of SL3(C) quantum trace map.
New spin on Khovanov-Rozansky homology categorifies spin link polynomial.
problem Categorify spin link polynomial using colored Khovanov-Rozansky homology.
method Equip Λ^n-colored sl_{2n} Khovanov-Rozansky homology with an involution.
result Categorifies spin-colored so_{2n+1} quantum link polynomial for n=1,2,3.
New invariants from quantum group theory for hyperbolic 3-manifolds.
problem Computing invariants for hyperbolic 3-manifolds with boundary.
method Using modular doubles of quantum s l ( 2 ; R ) \mathfrak{sl}(2;\mathbb R) sl ( 2 ; R ) and 6 j 6j 6 j -symbols. result Invariants decay exponentially with hyperbolic volume and 1-loop terms.
Restricts quantum representations of mapping class groups to integral coefficients.
problem Integrality of non-semisimple quantum representations of mapping class groups.
method Exhibits explicit bases of states spaces that span Z [ ζ ] \mathbb{Z}[ζ] Z [ ζ ] -lattices invariant under mapping class groups. result Restricts quantum representations to integral coefficients from Q ( ζ ) \mathbb{Q}(ζ) Q ( ζ ) to Z [ ζ ] \mathbb{Z}[ζ] Z [ ζ ] . Extends Lawrence's representations to integral U q s l ( 2 ) U_q \mathfrak{sl}(2) U q sl ( 2 ) Verma-modules and braid groups.
problem Integrating Lawrence's representations into U q s l ( 2 ) U_q \mathfrak{sl}(2) U q sl ( 2 ) Verma-modules and braid groups. method Defining homological operators and showing they provide a representation for U q s l ( 2 ) U_q \mathfrak{sl}(2) U q sl ( 2 ) , establishing isomorphisms and preserving key properties. result Recovering an integral version of Kohno's theorem for Verma-modules and braid group representations.
The paper proves a quantum modularity conjecture for 3-manifolds.
problem Quantum invariants of 3-manifolds at roots of unity.
method Formulates and proves a strong version of the conjecture for geometric 3-manifolds.
result The conjecture holds for Brieskorn homology spheres and some other examples.
The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.
problem Connecting quantum trace and UV-IR maps for surfaces and 3-manifolds.
method Analyzing compatibility under triangulation changes and using skein modules.
result Compatibility of quantum trace and UV-IR maps for surfaces and 3-manifolds.
Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.
problem Constructing and studying new invariants for 4D 2-handlebodies.
method Defining invariants for pairs ( W , ω ) (W,ω) ( W , ω ) , using unimodular ribbon Hopf coalgebras. result Decomposition formulas for original invariants in terms of refined ones.
We show that we can release the rigidity of the skew Howe duality process for s l n {\mathfrak sl}_n s l n knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine s l m {\mathfrak sl}_m s l m case, corresponding to looking at tan…
We study relationships between the restricted unrolled quantum group U ‾ q H ( s l 2 ) \overline{U}_q^H(\mathfrak{sl}_2) U q H ( sl 2 ) at 2 r 2r 2 r -th root of unity q = e π i / r , r ≥ 2 q=e^{πi/r}, r \geq 2 q = e π i / r , r ≥ 2 , and the singlet vertex operator algebra M ( r ) \mathcal M(r) M ( r ) . We use deformable families of modules to efficiently compute ( 1 , 1 ) (1, 1) ( 1 , 1 ) -tangle invariants colored with projecti…
Unified approach to deform Lie-Hamilton systems using Poisson-Hopf algebra.
problem Deforming Lie systems with quantum algebras.
method Poisson-Hopf algebra deformations applied to Lie-Hamilton systems.
result Unified approach to deformations of Lie-Hamilton systems on the real plane.
New method calculates Chern-Simons volume for 3-manifolds with surgery diagrams.
problem Computing Chern-Simons volume for 3-manifolds with torus boundaries and cusps.
method Direct computation from surgery diagrams using a log-decoration and quantum group coordinates.
result Direct computation of Chern-Simons volume for 3-manifolds with torus boundaries and cusps.
Homological model for quantum representations of mapping class groups.
problem Investigate linearity of mapping class groups using quantum representations.
method Homological action on configuration space with twisted coefficients.
result Identify subrepresentation equivalent to quantum s l 2 \mathfrak{sl}_2 sl 2 representation. Study of s p 4 \mathfrak{sp}_4 sp 4 -webs on surfaces, proving cluster algebra structure.
problem Understanding s p 4 \mathfrak{sp}_4 sp 4 -webs and their cluster algebra properties. method Introduced skein algebra and cluster structure, proved positivity.
result Proved S s p 4 , Σ Z q [ ∂ − 1 ] \mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q}[\partial^{-1}] S sp 4 , Σ Z q [ ∂ − 1 ] is a quantum cluster algebra. Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.
problem Developing topological field theories from non-semisimple quantum groups.
method Using the unrolled quantum group of o s p ( 1 ∣ 2 ) \mathfrak{osp}(1 \vert 2) osp ( 1∣2 ) and a relative modular structure on weight modules. result Establishes a connection between constructed invariants and physicists' Z ^ \widehat{Z} Z -invariants. Quantizes Chern-Simons invariant for tangle exteriors.
problem Geometric quantization of Chern-Simons invariant for tangles.
method Defining a sequence of invariants Z N ψ \mathcal{Z}_{N}^ψ Z N ψ using modules over quantum s l 2 \mathfrak{sl}_{2} sl 2 and holonomy R R R -matrices. result Directly recovers Chern-Simons invariant when N = 1 N = 1 N = 1 . Lectures introduce evaluation of SL(3) foams and link homology.
problem Categorification of quantum s l 3 \mathfrak{sl}_3 sl 3 web and link invariant. method Introduction and review of S L ( 3 ) \mathsf{SL}(3) SL ( 3 ) foams and their evaluation. result Categorification of Kuperberg quantum s l 3 \mathfrak{sl}_3 sl 3 web and link invariant. Using quantum skew-Howe duality, we study the category Rep ( g l ( m ∣ n ) ) \operatorname{Rep}(\mathfrak{gl}(m|n)) Rep ( gl ( m ∣ n )) of tensor products of exterior powers of the standard representation of U q ( g l ( m ∣ n ) ) U_q(\mathfrak{gl}(m|n)) U q ( gl ( m ∣ n )) , and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a ca…
For each finite dimensional, simple, complex Lie algebra g \mathfrak g g and each root of unity ξ ξ ξ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant τ M g ( ξ ) ∈ C τ_M^{\mathfrak g}(ξ)\in \mathbb C τ M g ( ξ ) ∈ C of oriented 3-manifolds M M M . In the present paper we construct an invariant…
Khovanov homology helps create quantum error-correcting codes.
problem Creating robust quantum error-correcting codes.
method Using Khovanov homology and its extensions to define and analyze quantum codes.
result New families of quantum codes with desirable properties.
Study unbounded s l 3 \mathfrak{sl}_3 sl 3 -laminations around punctures.
problem Classify and understand structures of s l 3 \mathfrak{sl}_3 sl 3 -laminations at punctures. method Relate to root data, classify signed webs, describe tropicalization, clarify relationships with other approaches.
result Clarify the relationship between s l 3 \mathfrak{sl}_3 sl 3 -laminations and other approaches. New invariants derived from Kauffman bracket for 3-manifolds.
problem Quantum invariants of 3-manifolds, especially non-semisimple ones.
method Combinatorial methods using Temperley-Lieb algebras and Kauffman bracket polynomials.
result Recovery of invariants from small quantum group of s l 2 \mathfrak{sl}_2 sl 2 . The paper calculates colored Jones polynomials for specific link configurations.
problem Computing colored Jones polynomials in general is difficult, but the paper provides explicit formulas.
method Uses Kuperberg's A 2 A_2 A 2 skein relation and one-row Young diagrams. result Derives the s l 3 \mathfrak{sl}_3 sl 3 tail of ( 2 , 2 m ) (2,2m) ( 2 , 2 m ) -torus links and false theta series. Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.
problem Calculating colored Jones polynomials for general oriented links is difficult.
method Using Kuperberg's linear skein theory, they focus on one-row polynomials for pretzel links.
result Existence of tails for specific pretzel knots' Jones polynomials is shown.
Computes Lie algebra structure constants using a graphical calculus.
problem Computing Lie algebra structure constants efficiently.
method Graphical calculus for classical invariant theory.
result Generalizes known methods for s l 2 \mathfrak{sl}_2 sl 2 to other Lie algebras.