arXiv research

Extend Kauffman bracket skein module to homology theory using Heegaard splittings

Paper disproves a theorem about Kauffman bracket skein module structure.

Paper compares skein modules to Kauffman bracket modules.

Paper defines new invariants for surface-links using graph diagrams and magmas.

The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler measure of the Kauffman bracket. Second we find a general form for the Kauffman bra…

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

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.

We show that the Kauffman bracket $[L]$ of a checkerboard colorable virtual link $L$ is an evaluation of the Bollobás-Riordan polynomial $R_{G_L}$ of a ribbon graph associated with $L$. This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs.

New invariants derived from Kauffman bracket for 3-manifolds.

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

Model proteins with bonds using Kauffman bracket skein module.

Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.

In the present paper, we build a bridge between Conway-Coxeter friezes and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomials attached to rational links by using Conway-Coxeter friezes. As an application one can give a complete invariant on Conway-Coxeter friezes o…

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 computes a knot's Kauffman bracket polynomial using recursive concatenation of a 4-tangle shadow.

Study Kauffman bracket skein modules of Seifert fibered spaces.

Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.

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

We provide an explicit, finite set of generators for the Kauffman bracket arc algebra defined by Roger and Yang.

In this paper, we formulate a construction of ideal coset invariants for surface-links in $4$-space using invariants for knots and links in $3$-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an invariant for surface-links called the Kauffman bracket ideal coset invariant of sur…

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…

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…

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…

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

Found a basis and presentation for a specific algebra.

We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A functorial characterization of this algebra is given. Differentiability properti…

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

This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn, Kauffman and Manturov.

For a ribbon graph $G$ we consider an alternating link $L_G$ in the 3-manifold $G\times I$ represented as the product of the oriented surface $G$ and the unit interval $I$. We show that the Kauffman bracket $[L_G]$ is an evaluation of the recently introduced Bollobas-Riordan polynomial $R_G$. This results generalizes t…

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

Paper reviews Viro's definition of Khovanov homology for tangles.

We provide a presentation of the Roger and Yang's Kauffman bracket arc algebra for the once-punctured torus and punctured spheres with three or fewer punctures.

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…

Kauffman bracket skein algebra structure on surfaces defined.

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

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.

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket skein module is a linear combination of $O(2^g)$ basis elements, with…

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.

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

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 …

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.

Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define …

Analog of Kauffman bracket for non-orientable knots in thickened surface.

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.

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.

Presented an algebra structure for a specific geometric surface.

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