The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.
problem Improving bounds on skein tree depth and delta-crossing numbers for knots and links.
method Theoretical and computational analysis of skein trees and knot invariants.
result New upper and lower bounds on skein tree depth and delta-crossing numbers are derived.
The paper improves bounds on the complexity of computing link polynomials.
problem Computing link polynomials by the skein relation is complex.
method Proved new upper and lower bounds on skein tree depth.
result New bounds on skein tree depth are stronger than previous ones.
Study on unimodality of plucking polynomial with delay function.
problem Exploring unimodality of plucking polynomial with delay function.
method Presented a formula for the plucking polynomial of hedgehog rooted trees and explored unimodality with specific delay functions.
result Found interesting examples and speculations on unimodality of plucking polynomials with delay functions.
We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over Z/2Z. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The (E2,d2) page of this spectral sequence …
We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree …
We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete re…
Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy pol…
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.
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.
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.
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.
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.
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…
Center identified in stated skein algebra for quantum traces.
problem Understanding the center of the stated skein algebra.
method Analyzing the algebra as a generalization of Kauffman bracket skein algebra, focusing on the case when the quantum parameter is a root of unity.
result Simple description and dimension calculation of the center over the center module.
Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
problem Exploring the Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
method Generalization of splitting homomorphism for stated skein modules of 3-manifolds.
result Existence and properties of Chebyshev-Frobenius homomorphism for 3-manifold skein modules.
Study of skein invariants on tori for various groups and quantum parameters.
problem Analysis of G-skein theory invariants on tori for different groups and parameters. method Combinatorial and algebraic methods, including DAHA and skein relations.
result Isomorphisms and homomorphisms between skein algebras and DAHA, proving equivalence of tangles.
Proves finiteness and holonomicity of skein modules for 3-manifolds.
problem Finiteness and holonomicity of skein modules for 3-manifolds.
method Defining skein transfer bimodules and using q-analogues of D-module theory.
result Internal skein modules are holonomic modules over the internal skein algebra of the boundary.
Decomposes SL3 skein algebras for surfaces.
problem Decomposing SL3 skein algebras for surfaces. method Splitting surfaces into triangles and analyzing the resulting algebras.
result Explicit basis and injective splitting morphisms for SL3 stated skein algebras. Propose a model-independent axiomatic framework for derived skein theory.
problem Derived skein theory of oriented 3-manifolds with coefficients in a ribbon tensor category.
method Design axioms for the 0th homology and gluing.
result Establishes relationships between derived and ordinary skein theory.
This paper calculates the skein algebra of the Borromean rings complement.
problem Calculating the skein algebra of the Borromean rings complement.
method Using the skein algebra definition and character variety, the polynomial ring quotient is determined.
result An explicit formula for the skein algebra of the Borromean rings complement is provided.
Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpret…
New skein categories for non-semisimple settings, extending existing theory.
problem Extending skein theory to non-semisimple settings.
method Introducing skein categories based on tensor ideals in linear ribbon categories.
result Skein categories coincide with factorization homology in non-semisimple settings.
Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.
problem Understanding the genus-two handlebody skein module.
method Using q-difference operators for the genus-two skein algebra.
result Correspondence between reduced Askey-Wilson polynomials and genus-two handlebody skein module.
Introduces admissible skein modules for non-semisimple categories.
problem No specific problem stated; generalization of Kauffman skein algebra.
method Introduces admissible skein modules associated to ideals in pivotal categories.
result These modules generalize Kauffman skein algebra and relate to quantum invariants.
Proves a pentagon relation in skein theory.
problem Developed a skein-theoretic version of cluster theory and conjectured a pentagon relation.
method Topological proof using skein algebra and elliptic Hall algebra.
result Proves the pentagon relation for the skein dilogarithm.
Survey of stated skein modules/algebras of 3-manifolds/surfaces.
problem Understanding stated skein modules/algebras of 3-manifolds/surfaces.
method Discussion of splitting homomorphism, general structures, Frobenius homomorphism, center, dimension, representation theory.
result Skein algebra of non-closed marked surface at any root of 1 is a maximal order.
Study centers of generalized skein algebras, showing almost Azumaya properties.
problem Understanding the center of generalized skein algebras.
method Generalized skein algebra generated by loops and arcs, computed center, discussed implications.
result Center of Muller-Roger-Yang skein algebra is almost Azumaya.
Researchers compute the Kauffman bracket skein module of a specific 3-manifold.
problem Understanding the structure of Kauffman bracket skein modules for non-prime manifolds.
method Analyzing handle sliding relations to compute the module over Z[A±1]. result The skein module of (S1imesS2) # (S1imesS2) does not split into free and torsion submodules. New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.
problem Understanding relationships between Frobenius elements and Jones-Wenzl projectors at roots of unity.
method Obtained skein identities relating Frobenius elements to Jones-Wenzl projectors in the Kauffman bracket skein module.
result Skein identities provide new proofs of the existence of the Chebyshev-Frobenius homomorphism.
A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …
Proves a new skein exact triangle for real monopole Floer homology.
problem None explicitly stated; focuses on proving a new mathematical structure.
method Introduces a new exact triangle for real monopole Floer homology.
result Proves an unoriented skein exact triangle for real monopole Floer homology.
Study of cubic skein modules in 3-sphere and arbitrary 3-manifolds.
problem Lack of systematic study of higher degree skein modules.
method Investigation of cubic skein module structure and properties in 3-sphere and arbitrary 3-manifolds.
result Establishment of a foundational framework for higher skein modules.
Study Morse theory on loop spaces and Hecke algebras.
problem Morse theory applied to loop spaces and Hecke algebras.
method Defined a Morse-type A∞-algebra and showed equivalence to Heegaard Floer algebras. result Equivalence of based multiloop A∞-algebra to wrapped higher-dimensional Heegaard Floer algebras. New skein exact triangles for link Floer homology.
problem Understanding link Floer homology through skein relations.
method Construction of skein triples for rational tangles.
result Established a framework for potential further skein exact triangles.
Skein algebra action is faithful if quantum parameter isn't a root of 1.
problem Faithfulness of geometric action of skein algebras.
method Examined the Kauffman bracket skein algebra and its action on handlebody modules.
result The action is faithful if and only if the quantum parameter is not a root of 1.
Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.
problem Studying representations of Kauffman bracket skein algebras at roots of unity.
method Using the action of the skein algebra on the skein module of the handlebody.
result Explicit reconstruction of unique representation with fixed classical shadow.
Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.
problem Study sliced skein algebras and their properties.
method Quotient of Kauffman bracket skein algebra, center calculation, PI-degree calculation, fully Azumaya point analysis.
result Center and PI-degree calculations for sliced skein algebras, fully Azumaya points, and simple modules.
We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming para…
For a Catalan state C of a lattice crossing L(m,n) with no returns on one side, we find its coefficient C(A) in the Relative Kauffman Bracket Skein Module expansion of L(m,n). We show, in particular, that C(A) can be found using the plucking polynomial of a …
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.
Introduces LRY skein algebras generalizing Kauffman bracket and Roger-Yang skein algebras.
problem Generalizing skein algebras for surfaces with arbitrary ground rings.
method Constructs LRY skein algebras, quantum traces, and Dehn-Thurston coordinates.
result LRY skein algebras are domains, have degenerations to monomial subalgebras of quantum tori, and are orderly finitely generated.
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. Study G2 skein algebra elements using Kuperberg webs and threading operations.
problem Identify and analyze central elements in G2 skein algebra. method Use Kuperberg webs, threading operations, and skein-theoretic arguments.
result Verify central elements and obtain uniqueness of transparent polynomials.
Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.
problem Positivity conjecture for Roger--Yang skein algebras.
method Used explicit polynomials like Chebyshev polynomials of the first kind to give candidates of positive bases.
result Polynomials form a lower bound in the sense of [Lê18] and [LTY21].
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives …
Extend Kauffman bracket skein module to homology theory using Heegaard splittings
problem Extend Kauffman bracket skein module to homology theory
method Combinatorial approach using Heegaard splittings
result Homology theory depends on Heegaard splittings
We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.