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…
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48 results for “Relative Kauffman Bracket Skein Module”
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.
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
Paper disproves a theorem about Kauffman bracket skein module structure.
problem Disproving a 22-year-old theorem about Kauffman bracket skein module structure.
method Analyzing handle slidings on compressing discs in handlebodies.
result More relations found than previously predicted for connected sum of handlebodies.
Formula for Dehn twists in skein algebras defined on surfaces.
problem Calculating the effect of Dehn twists on skein algebras.
method Introduced filtrations of Kauffman bracket skein algebra and modules.
result Explicit formula for Dehn twists action.
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 for ${
m KBSM}\left({
m ST}
ight)$.
New generators identified for 3-torus skein module.
problem Identifying generators for Kauffman bracket skein module of 3-torus.
method Showed linear independence of 9 identified generators.
result Linearly independent set of generators for 3-torus skein module.
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 .
result The skein module of does not split into free and torsion submodules.
We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein modu…
The paper studies Kauffman bracket skein modules of lens spaces.
problem Investigating Kauffman bracket skein modules of lens spaces.
method Representing links through band and disk diagrams, transforming between them, and computing the module for specific examples.
result The Kauffman bracket skein module is an essential invariant, showing different values on inequivalent links with equivalent lifts.
Proves finiteness of Kauffman bracket skein modules for 3-manifolds.
problem Finiteness of Kauffman bracket skein modules for 3-manifolds.
method Constructive proof methods, alternative to Witten's original proof.
result Proves finiteness conjecture for 3-manifolds with boundary and closed 3-manifolds.
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.
New embedding connects Torelli group to skein module.
problem Understanding Torelli group structure.
method Embedding into Kauffman bracket skein algebra.
result New construction of Johnson homomorphism.
Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.
problem Computing Kauffman bracket skein modules of Seifert manifolds.
method Provided presentations to compute the modules and showed their properties.
result The modules of certain Seifert manifolds are free or finitely generated.
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.
Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.
problem Determining the structure of Kauffman bracket skein module of connected sums of solid tori.
method Used algebraic methods over the ring of Laurent polynomials to prove the conjecture.
result Proved a conjecture about the Kauffman bracket skein module of connected sums of genus one handlebodies.
Study the algebraic action of torus on knot complement's skein module.
problem Understand the algebraic structure of knot complements and boundary tori.
method Analyze the Kauffman bracket skein algebra and module of the 3-twist knot complement.
result Determine the action of Kauffman bracket skein algebra on module of 3-twist knot complement.
Study on skein modules and character varieties of Seifert manifolds.
problem Characterizing and analyzing skein modules and character varieties of Seifert manifolds.
method Analyzing character varieties and skein modules of Seifert manifolds, computing dimensions and showing their equivalence.
result Finitely generated skein modules and reduced character varieties for certain Seifert manifolds.
We reprove and expand results of Bonahon and Wong on central elements of the Kauffman bracket skein modules at root of 1 and on the existence of the Chebyshev homomorphism, using elementary skein methods.
Study disproves finiteness conjecture for a specific link's skein module.
problem Finiteness conjecture for a specific link's skein module.
method Analysis of Kauffman bracket skein module over .
result Skein module is not finitely generated over .
New bases found for Kauffman bracket skein module of fibered torus.
problem Computing Kauffman bracket skein module of fibered 3-manifolds.
method Constructed bases for Kauffman bracket skein module of product annulus and circle, then applied to -fibered torus.
result Found a new basis for KBSM of -fibered torus.
Model proteins with bonds using Kauffman bracket skein module.
problem Modeling proteins with bonds for structural analysis.
method Extend Kauffman bracket polynomial to bonded knots.
result Infinite generation and torsion-freeness of the bonded skein module.
We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds.
Study shows skein module dimensions for surface times circle.
problem Understanding skein modules of surface times circle.
method Used Kauffman bracket skein module over rational functions.
result Dimension at least 2^{2g+1}+2g-1 for Sigma x S^1.
We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, …
We show that the Kauffman bracket skein modules of certain manifolds obtained from integral surgery on a (2,2b) torus link are finitely generated, and list the generators for select examples.
Counterexample disproves conjecture about 3-manifold modules.
problem Disproving a conjecture about 3-manifold modules.
method Provided a specific counterexample.
result Disproved Marché's conjecture about 3-manifold modules.
The paper proves unicity for Kauffman bracket skein algebras.
problem Unicity conjecture for Kauffman bracket skein algebras of surfaces.
method General unicity theorem applied to Kauffman bracket skein algebras, center characterization, and finitely generated module over center.
result Irreducible representations of Kauffman bracket skein algebras are classified by their central characters.
Novel method computes Kauffman bracket skein modules of small 3-manifolds.
problem Computing Kauffman bracket skein modules of small 3-manifolds.
method Developed a novel method to compute these modules.
result Dimension of skein modules of 3-manifolds related to character varieties and Abouzaid-Manolescu invariants.
This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and o…
New findings show the Gilmer-Masbaum map isn't always one-to-one.
problem Determining the injectivity of the Gilmer-Masbaum map on Kauffman bracket skein modules.
method Computed the image of the evaluation map for specific cases of mapping tori and analyzed homology classes.
result The restriction of the Gilmer-Masbaum map to certain homology classes is not injective.
Constructs maps on skein modules using non-semisimple quantum invariants.
problem Constructing maps on skein modules with specific characters.
method Uses non-semisimple invariants of 3-manifolds.
result Maps with any possible abelian non-central character as classical shadow.
New bases for Kauffman bracket skein module of genus 2 handlebody.
problem Finding new bases for Kauffman bracket skein module of genus 2 handlebody.
method Using parting technique to convert elements in the Przytycki-basis to open braid form, defining an ordering relation, and relating the bases via matrix relations.
result Introducing as a more natural basis for KBSM() and suitable for computing modules of 3-manifolds obtained from by surgery.
This paper extends Khovanov homology to categorify Kauffman bracket skein module for non-orientable surface.
problem Categorify Kauffman bracket skein module for non-orientable surface .
method Redefined the differential in the Khovanov chain complex for the twisted -bundle over .
result Categorified the Kauffman bracket skein module for the non-orientable surface .
New method finds coefficients of Catalan states using plucking polynomials.
problem Finding coefficients of Catalan states in lattice crossings.
method Using plucking polynomials of rooted trees with delay functions.
result Coefficients of Catalan states are unimodal sequences.
Researchers calculate dimensions of skein modules for 2-torus mapping tori.
problem Determining dimensions of Kauffman bracket skein modules for specific cases.
method Using generic and decomposing twisted Hochschild homology of -skein algebras.
result Dimensions of skein modules for and are calculated.
Invariants of multi-linkoidsmath.GT
Extends knotoid theory to multi-linkoids, studying various invariants.
problem Defining and analyzing invariants for multi-linkoids in a closed surface.
method Examines Kauffman bracket, ordered bracket, skein module, and T-invariant.
result Developed new invariants for multi-linkoids.
Study torsion in Kauffman bracket skein modules of 3-manifolds.
problem Identify compact oriented 3-manifolds with torsion in their Kauffman bracket skein modules.
method New criteria based on -character varieties and effective criteria for manifolds with incompressible tori.
result Many counterexamples to Kirby's list and new criteria for torsion presence.
Study shows torsion in knot module for specific Montesinos knots.
problem Torsion in Kauffman bracket skein module of Montesinos knots.
method Analyzes Kauffman bracket skein module over for specific knots.
result Provides a negative answer to Problem 1.92 in Kirby's list.
This thesis explores DAHA representations using stated skein theory.
problem Understanding the representation theory of double affine Hecke algebras.
method Combining stated skein theory with DAHA, focusing on the DAHA.
result Constructed a module of Laurent polynomials for the DAHA.
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.
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 provided for any closed oriented surface .
result The dimension of is .
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.
We calculate the Kauffman bracket skein module (KBSM) of the complement of all two-bridge links. For a two-bridge link, we show that the KBSM of its complement is free over the ring $\BC[t^{\pm 1}]$ and when reducing , it is isomorphic to the ring of regular functions on the character variety of the link group.
We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a…
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 compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other co…
For each closed, orientable surface F, we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module K_t(F x [0,1]). The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = - 1, the trace is integration against the symplectic measure on the SU(2) character var…