Computes Kauffman bracket polynomial for specific 2-tangle shadows.
problem Calculating Kauffman bracket polynomial for complex tangle structures.
method Computed Kauffman bracket polynomial for specific 2-tangle shadows with up to 4 crossings.
result Computed polynomial for specific 2-tangle shadows.
The Kauffman bracket polynomial is calculated for specific Turk's head knots.
problem Computing the Kauffman bracket polynomial for Turk's head knots.
method The 3-tangle is repeatedly concatenated and then closed. State diagrams are expressed using the Kauffman monoid diagram elements.
result The Kauffman bracket polynomial values for the three-lead Turk's head, chain sinnet, and figure-eight chain shadow are computed.
We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás-Riordan polynomial RGL 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.
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…
The paper computes a knot's Kauffman bracket polynomial using recursive concatenation of a 4-tangle shadow.
problem Computing the Kauffman bracket polynomial for complex knots.
method Recursive concatenation of a 4-tangle shadow, followed by a closure operation and polynomial computation.
result A method to compute the Kauffman bracket polynomial for knots formed from 4-tangle shadows.
Paper defines new invariants for surface-links using graph diagrams and magmas.
problem Tackles invariants for surface-links in entropic magmas.
method Uses marked graph diagrams and a generalization of Kauffman bracket magma.
result Defines new invariants for surface-links in 4-space.
Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.
problem Calculating the Kauffman bracket polynomial for Celtic link shadows.
method Two complementary approaches: recursive relation and 4-tangle algebra.
result Derived Kauffman bracket polynomial for CK42n shadows. 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.
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…
For a ribbon graph G we consider an alternating link LG in the 3-manifold G×I represented as the product of the oriented surface G and the unit interval I. We show that the Kauffman bracket [LG] is an evaluation of the recently introduced Bollobas-Riordan polynomial RG. This results generalizes t…
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.
Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant tI(L) is constructed for a link L, where I is the abelian Chern-Simons action and t a formal constant. For oriented knotted vortex lines, tI satisf…
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.
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.
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 revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.
problem Understanding polynomial invariants of pretzel links.
method Revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links P(1,1,n). result Reveals properties of Alexander-Conway and Kauffman bracket polynomials for P(1,1,n). Dye and Kauffman defined surface bracket polynomials for virtual links by use of surface states, and found a relationship between the surface states and the minimal genus of a surface in which a virtual link diagram is realized. They and Miyazawa independently defined a multivariable polynomial invariant of virtual lin…
Analog of Kauffman bracket for non-orientable knots in thickened surface.
problem Defining an invariant for non-orientable knots in a non-orientable surface.
method Proposes an analog of the Kauffman bracket polynomial with modified sign rules.
result Polynomial is an isotopy invariant and independent of classical Kauffman for orientable covers.
We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials…
We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.
The Kauffman-Vogel polynomials are three variable polynomial invariants of 4-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented 4-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with 2. Bataineh, Elha…
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.
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.
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(2g) basis elements, with…
Let G be a signed graph. Let G^ be the graph obtained from G by replacing each edge e by a chain or a sheaf. We first establish a relation between the Q-polynomial of G^[6] and the W-polynomial of G [9]. Two special dual cases are derived from the relation, one of which has been studied in [8]…
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…
New invariants derived from Kauffman bracket for 3-manifolds.
problem Quantum invariants of 3-manifolds, especially non-semisimple ones.
method Combinatorial methods using Temperley-Lieb algebras and Kauffman bracket polynomials.
result Recovery of invariants from small quantum group of sl2. 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 invariant for tied links connects states without resolution dependence.
problem Understanding the Kauffman-like states for tied links.
method Defined Aicardi-Juyumaya states and showed their contribution to the invariant is independent of resolution.
result The double bracket of a tied link diagram can be computed and used to find linked but differently polynomial tied links.
Algorithm calculates Jones polynomial from Goeritz matrix.
problem Calculating Jones polynomial from link diagrams.
method Explicit algorithm using Goeritz matrices.
result Jones polynomial can be recovered from orientable checkerboard surfaces.
Computes Jones polynomial for specific knots.
problem Computing Jones polynomial for double twist knots.
method Using cyclotomic expansion and Kauffman bracket skein theory.
result Answers a question about Jones polynomial for specific knots.
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 RP2. method Redefined the differential in the Khovanov chain complex for the twisted I-bundle over RP2. result Categorified the Kauffman bracket skein module for the non-orientable surface RP2. 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…
New link polynomials linked to cluster theory.
problem Connecting link polynomials to cluster theory.
method Introducing new link polynomials and their expansion over perfect matchings.
result Bracket polynomials of certain links can be realized as specializations of cluster variables.
Study links in 3-manifolds, linking volume to polynomial coefficients.
problem Understanding the volume of hyperbolic links in 3-manifolds.
method Using Kauffman bracket functions and polynomial invariants, linking volume to polynomial coefficients.
result Coefficients of polynomial provide 2-sided linear bounds on the volume of hyperbolic links.
Given any diagram of a link, we define on the cube of Kauffman's states a "2-complex" whose homology is an invariant of the associated framed links, and such that the graded Euler characteristic reproduces the unnormalized Kauffman bracket. This includes a categorification of brackets skein relation. Then we incorporat…
Quantum approach to volume computation from colored Jones polynomials.
problem Computing volumes of hyperbolic 3-manifolds from knot polynomials.
method Categorification of Jones polynomials and asymptotic analysis of skein elements.
result Asymptotic growth rate of Kauffman bracket relates to volumes of ideal octahedra.
This paper explains how to compute Khovanov homology of torus links using the Kauffman bracket polynomial.
problem Computing Khovanov homology of specific knot types.
method Categorification of the Kauffman bracket polynomial via a long exact sequence.
result A practical method to compute Khovanov homology of torus links.
Proof of isotopy invariance in Khovanov link homology.
problem Isotopy invariance of Khovanov link homology.
method Explicit description of retractions and chain homotopies.
result Isotopy invariance of Khovanov link homology proved.
This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. The paper is relatively self-contained and it describes virtual knot theory bot…
Bounding twist number of surface links using polynomial coefficients.
problem Bounding the twist number of alternating surface links.
method Introducing a generalized homological Kauffman bracket and applying it to surface link diagrams.
result A bound for the twist number of alternating surface links in terms of polynomial coefficients.
New invariant for virtual links defined using homology.
problem Invariants for virtual links and their properties.
method Defining homological arrow polynomial and using it to study virtual links.
result New invariant for virtual links and its properties.
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. S. Nelson, M. Orrison, V. Rivera {\cite{S}} modified Kauffman's construction of bracket. Their invariant ΦXβ takes value in a finite ring Z2[t]/(1+t+t3). In this paper, the author generalizes this invariant. The new invariant takes value in a polynomial ring. Furthermore, for a tricolorable link diagram, the au…
Paper introduces flat-virtual knots and invariants for classical knots.
problem Constructing a map from classical knots to virtual knots.
method Definition of flat-virtual knots and invariants (Alexander-like polynomial, Kauffman bracket).
result Introduction of flat-virtual knots and their invariants.
The paper extends knotoid theory to annular and toroidal settings.
problem Extending knotoid theory to new geometric settings.
method Introducing new knotoid types, extending bracket polynomials.
result Universal bracket polynomials for annular and toroidal knotoids.
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.
The paper extends knot polynomials to annular and toroidal pseudo links.
problem Analyzing pseudo links with undefined crossings.
method Introducing Kauffman bracket and Jones-type polynomials for annular and toroidal pseudo links.
result New tools for studying annular and toroidal pseudo links.