arXiv research

Extend Kauffman bracket skein module to homology theory using Heegaard splittings

Paper compares skein modules to Kauffman bracket modules.

Study the algebraic action of torus on knot complement's skein module.

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…

Researchers compute the Kauffman bracket skein module of a specific 3-manifold.

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory.

New bases found for Kauffman bracket skein module of fibered torus.

Paper disproves a theorem about Kauffman bracket skein module structure.

This thesis explores DAHA representations using stated skein theory.

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 give an explicit formula for the action of the Dehn twist along a simple closed curve in a compact connected oriented surface on the completion of the filtered skein modules. To do this, we introduce filtrations of the Kauffman bracket skein algebra and the Kauffman bracket skein modules on the surface.

Proves finiteness of Kauffman bracket skein modules for 3-manifolds.

Extends knotoid theory to multi-linkoids, studying various invariants.

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…

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant $t^{I\left( \mathcal{L} \right) }$ is constructed for a link $\mathcal{L}$, where $I$ is the abelian Chern-Simons action and $t$ a formal constant. For oriented knotted vortex lines, $t^{I}$ satisf…

Study Kauffman bracket skein modules of Seifert fibered spaces.

Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.

Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.

Computes Jones polynomial for specific knots.

Extends Kauffman's formula to 3-manifolds with markings.

New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.

We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.

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.

Quantized Coulomb branches linked to skein algebras.

Found a basis and presentation for a specific algebra.

The paper develops a theory of skein adequate links in thickened surfaces and proves Tait conjectures.

Carrega has shown that the Kauffman bracket skein module of the 3-torus over the field of rational functions in the variable A can be generated by 9 skein elements. We show this set of generators is linearly independent.

In this paper we give an alternative basis, $\mathcal{B}_{\rm ST}$, for the Kauffman bracket skein module of the solid torus, ${\rm KBSM}\left({\rm ST}\right)$. The basis $\mathcal{B}_{\rm ST}$ is obtained with the use of the Tempereley--Lieb algebra of type B and it is appropriate for computing the Kauffman bracket sk…

Model proteins with bonds using Kauffman bracket skein module.

Presented an algebra structure for a specific geometric surface.

Kauffman bracket skein algebra structure on surfaces defined.

Study on skein modules and character varieties of Seifert manifolds.

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…

Study disproves finiteness conjecture for a specific link's skein module.

For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link…

Let $F$ be a finite type surface and $ζ$ a complex root of unity. The Kauffman bracket skein algebra $K_ζ(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Fie…

The paper extends knotoid theory to annular and toroidal settings.

Skein algebra action is faithful if quantum parameter isn't a root of 1.

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.

Quantum approach to volume computation from colored Jones polynomials.

We introduce an embedding of the Torelli group of a compact connected oriented surface with non-empty connected boundary into the completed Kauffman bracket skein algebra of the surface, which gives a new construction of the first Johnson homomorphism.

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.

A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We show how an irreducible representation of the skein algebra usually specifies a …

The paper presents a new algebraic structure for planar surfaces.

We give an explicit basis $\mathcal{B}$ of the quotient of the Kauffman bracket skein algebra $\mathcal{S} (Σ)$ on a surface $Σ$ by the square of an augmentation ideal. As an application, it induces two kinds of finite type invariants of links in a handle body in the sense of Le. Moreover, we construct an embedding of …

Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.

New findings show the Gilmer-Masbaum map isn't always one-to-one.

Counterexample disproves conjecture about 3-manifold modules.