Paper presents skein algebras for spheres with punctures.
problem Quantization of decorated Teichmüller space.
method Presentations of Roger-Yang generalized skein algebras for punctured spheres.
result New interpretation of homogeneous coordinate ring of Grassmannian of planes.
Researchers compute gl2-skein modules for lens spaces.
problem Computing gl2-skein modules for lens spaces. method Action of gl2-skein algebra on solid torus's gl2-skein module. result Lens spaces' gl2-skein modules span by specific elements. The paper calculates colored Jones polynomials for 2-bridge links using skein theory.
problem Calculating colored Jones polynomials for 2-bridge links.
method Graphical calculus and skein theory.
result Explicit calculation of sl2 and sl3 colored Jones polynomials for 2-bridge links. Constructs a trace invariant for 3-manifold skein spaces.
problem Invariants for 3-manifold skein spaces.
method Local diffeomorphism invariant trace construction.
result Local diffeomorphism invariant trace on skein space.
Study Kauffman bracket skein modules of Seifert fibered spaces.
problem Understanding the structure of Kauffman bracket skein modules.
method Investigate spanning sets and module structure.
result Kauffman bracket skein modules are finitely generated.
9 generators found for 3-torus skein space.
problem Identifying generators for the skein space of the 3-torus.
method Analyzing the first homology group and constructing specific knots and links.
result The skein space of the 3-torus is generated by 9 specific elements.
Paper computes skein modules of 3-manifolds using braids.
problem Computing skein modules of 3-manifolds.
method Using braids and knot algebras.
result Braid approach to HOMFLYPT and Kauffman bracket skein modules of specific 3-manifolds.
Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.
problem Understanding the skein-valued cluster transformation in Legendrian surfaces.
method Geometric considerations of moduli of holomorphic curves.
result Skein-valued cluster transformation of Legendrian surfaces related by surgery.
Proves method to compute HOMFLYPT skein module of lens spaces L(p,1) from solid torus.
problem Computing HOMFLYPT skein module of lens spaces L(p,1) from solid torus. method Solving infinite system of equations involving Lambropoulou invariant and braid band moves.
result Derives HOMFLYPT skein module of lens spaces L(p,1) from solid torus. Alternative basis for Kauffman bracket skein module of solid torus found using braids.
problem Computing Kauffman bracket skein module of lens spaces.
method Using Temperley--Lieb algebra of type B and braids.
result Alternative basis BmST for ${
m KBSM}\left({
m ST}
ight)$. Injective homomorphism proves no zero divisors in Roger-Yang skein algebra.
problem Injectivity of Roger-Yang's homomorphism and zero divisors in skein algebra.
method Hyperbolic geometric considerations, ideal triangulation, normal arcs.
result Injective homomorphism proves no zero divisors in Roger-Yang skein algebra.
A map from 3-manifold skein to Lagrangian skein via holomorphic curve counting.
problem Counting holomorphic curves in cotangent bundles for 3-manifold skein.
method Skein-valued counting of holomorphic curves in branched covers.
result Wall-crossing formula for skein traces in branched covers.
Proves conjecture linking cluster algebras and skein algebras for surfaces with punctures.
problem Cluster and skein algebras on surfaces with punctures.
method Geometric and algebraic methods, including decorated Teichmüller spaces and skein algebras.
result Cluster and skein algebras coincide for surfaces with at least 2 punctures.
Homflypt skein theory and string topology linked via 2-groupoids.
problem Understanding relations in Homflypt skein theory.
method Defined a 2-groupoid from the fundamental 2-groupoid of singular links, relating it to string topology.
result Relations in Homflypt skein theory are induced from a 2-groupoid.
Computational study of prime knots in specific spaces.
problem Tabulation and classification of prime knots in lens spaces.
method Computational techniques, skein modules, hyperbolic structures.
result Established amphichiral knots and distinguished knots by skein modules.
Skein lasagna module calculates 4-manifold invariants using handle decompositions.
problem Calculating invariants of 4-manifolds and links.
method Express skein lasagna module in terms of handle decompositions.
result Skein lasagna module can be locally infinite dimensional.
The paper extends asymptotic faithfulness to skein quantum representations of mapping class groups.
problem Lack of explicit fusion spaces with multiplicities for skein quantum representations.
method Generalizes asymptotic faithfulness to skein quantum representations of mapping class groups.
result Conjectures asymptotic faithfulness for skein quantum G representations when G is a simply-connected simple Lie group. Survey on link diagrams in Seifert manifolds and skein modules.
problem Understanding link invariants in Seifert manifolds.
method Use of arrow diagrams and Reidemeister moves.
result New bases of skein modules for specific Seifert manifolds.
We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.
Quantum traces embed into quantum tori for surface skein algebras.
problem Embedding stated skein algebras into quantum tori.
method Two different embeddings using quantum trace maps and lambda length coordinates.
result Quantum cluster algebra of Muller equals reduced stated skein algebra.
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. Quantum torus methods enhance understanding of skein algebras and modules.
problem Investigating Kauffman bracket skein algebras and modules using quantum torus methods.
method Two quantum torus methods: embedding into quantum Teichmüller space and filtering to a monomial subalgebra.
result Generalized Chebyshev homomorphism and refined unicity theorem for skein modules.
Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.
problem Understanding transverse link invariants in the annular setting.
method Constructs a stable homotopy type for annular links and defines a map to the Khovanov skein spectrum.
result At extreme gradings, the map from the Khovanov spectrum to the Khovanov skein spectrum recovers the cohomotopy transverse invariant.
Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.
problem Understanding BPS q-series for 3-manifolds with line defects. method Proving homomorphism from skein module to space of q-series, conjecturing holomorphic modularity. result Holomorphic quantum modularity of q-series suggests new approach to Langlands duality. We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract …
Study skein modules via gauge theory, finding non-TQFT dimensions.
problem Understanding skein modules of 3-manifolds using gauge theory.
method Embed skein modules into 4d N=4 super-Yang-Mills theories, using N=1 supersymmetry. result Find non-standard dimensions of skein modules, differing from TQFT predictions.
Skein modules over 3-manifolds are shown to form line bundles.
problem Understanding the structure of skein modules over 3-manifolds.
method Using the Frobenius morphism, the skein module is mapped to a coherent sheaf over the SL2 character scheme.
result When the character scheme is reduced, the sheaf is a line bundle.
The study provides a basis for a 3-manifold's skein module, answering its dimension.
problem Determining the dimension of the Kauffman skein module of a surface times a circle.
method Explicitly spanning family for the skein modules S(ΣimesS1) provided for any closed oriented surface Σ. result The dimension of S(ΣimesS1) is 22g+1+2g−1. Researchers compute the rank and trace of Kauffman bracket skein algebra over its center.
problem Computing the rank and trace of Kauffman bracket skein algebra.
method Using finite type surfaces and complex roots of unity, they compute the rank and trace of Kζ(F) over its center. result They extend a theorem about the skein algebra having a splitting coming from two pants decompositions of F. New representations of pure braid groups defined from A2 web spaces.
problem Understanding the structure of pure braid groups.
method Defining representations of pure braid groups using A2 web spaces and calculating matrix representations. result Matrix representations of ρn about the standard generators of P2k calculated. The paper studies algebraic and geometric properties of stated skein algebras of surfaces.
problem Understanding the algebraic and geometric properties of stated skein algebras of surfaces.
method Analyzes the skein algebra of surfaces, proving isomorphisms and lifting properties, and interpreting topologically.
result The skein algebra of a surface with n boundary components is an algebra-comodule over Oq2(SL(2))⊗n. New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.
problem Proving the structure of SL(n) skein algebra for a specific surface.
method Constructing a linear basis of explicit SL(n) webs, proving spanning and linear independence.
result SL(n) skein algebra of twice punctured sphere is a commutative polynomial algebra in n-1 generators.
Alexander polynomial linked to 3-sphere's in lens spaces.
problem Alexander polynomial of links in lens spaces.
method Relationships between Alexander polynomials in 3-sphere and lens spaces.
result Normalization of Alexander polynomial satisfies skein relation in lens spaces.
The paper explores basic properties of knot skein invariants.
problem Understanding skein invariants of knots.
method Discussion of basic properties and known examples of skein invariants.
result Discussion of basic properties and known examples of skein invariants.
Extends Kauffman's formula to 3-manifolds with markings.
problem Generalizing Jones polynomial to 3-manifolds with markings.
method Epimorphism between skein modules of tangles in 3-manifolds.
result Defines new skein modules measuring the difference between Jones and bracket skein modules.
In this paper the properties of the Kauffman bracket skein module of L(p,q) are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the Kauffman bracket skein module on an interesting class of examples consisti…
Researchers compute HOMFLYPT skein module of lens spaces using braids.
problem Computing the HOMFLYPT skein module of lens spaces L(p,1). method Using braids to solve an infinite system of equations.
result Established the relation between S(L(p,1)) and S(ST). Study resolves conjecture linking two algebraic structures on surfaces.
problem Compatibility of skein and cluster algebra structures on surfaces.
method Established compatibility between skein and cluster algebras of surfaces.
result Cluster algebra of positive genus surfaces is not finitely generated.
Paper compares skein modules to Kauffman bracket modules.
problem Comparing skein modules to Kauffman bracket modules.
method Using skein relations and Reshetikhin-Turaev model.
result Resolved the problem of comparing skein modules to Kauffman bracket modules.
New findings on hyperbolic 3-manifolds with finite skein modules.
problem Finite dimensional skein modules of hyperbolic 3-manifolds.
method Introduced and proved stability property for Dehn-filling of knots.
result Almost all Dehn-fillings of two-bridge knots have finite skein modules.
Diagrams and Reidemeister moves for links in a twisted S^1-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of the connected sum of two projective spaces. In particular, we show that it has torsion. We also present a new computation of the KBS…
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.
Relates two types of skein algebras using explicit correspondences.
problem Defining and relating stated and internal skein algebras.
method Explicit correspondence between stated and internal skein algebras, distinguishing between left and right boundary edges, proving excision properties.
result Agrees with excision properties of stated skein algebras under specific conditions.
This paper computes the Homflypt skein module of lens spaces using braids.
problem Computing the Homflypt skein module of lens spaces L(p,1). method Using Lambropoulou invariant and braid band moves on a basis of the solid torus.
result Established the connection and computed the Homflypt skein module of lens spaces.
Survey of knot invariants in lens spaces.
problem Comparing knot invariants in lens spaces.
method Summarized various invariants and compared their distinguishing power.
result Invariants generalize classical knot polynomials and can distinguish difficult cases.
For a surface F, the Kauffman bracket skein module of F×[0,1], denoted K(F), admits a natural multiplication which makes it an algebra. When specialized at a complex number t, nonzero and not a root of unity, we have Kt(F), a vector space over C. In this paper, we will use the product-to-su…
Survey on stated skein algebras and their representations.
problem None explicitly stated in the abstract.
method None explicitly stated in the abstract.
result None explicitly stated in the abstract.
Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…