Kauffman Bracket Research Articles

Knotted 4-regular graphs: Polynomial invariants and the Pachner moves

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs—one inspired by the Kauffman bracket relations and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied.

Quantum traces and embeddings of stated skein algebras into quantum tori

The stated skein algebra of a punctured bordered surface (or equivalently, a marked surface) is a generalization of the well-known Kauffman bracket skein algebra of unmarked surfaces and can be considered as an extension of the quantum special linear group \({\mathcal {O}}_{q^2}(SL_2)\) from a bigon to general surfaces. We show that the stated skein algebra of a punctured bordered surface with non-empty boundary can be embedded into quantum tori in two different ways. The first embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the shear coordinates of the enhanced Teichmüller space, and is a lift of Bonahon-Wong’s quantum trace map. The second embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the lambda length coordinates of the decorated Teichmüller space, and is an extension of Muller’s quantum trace map. We explain the relation between the two quantum trace maps. We also show that the quantum cluster algebra of Muller is equal to a reduced version of the stated skein algebra. As applications we show that the stated skein algebra is an orderly finitely generated Noetherian domain and calculate its Gelfand-Kirillov dimension.

Embedding tangles and the Kauffman bracket polynomials

We give a necessary condition for a 4-tangle T to be embedded in a link by using a common divisor of two integers obtained from the Kauffman bracket polynomials of the numerator N(T) and the denominator D(T) of the tangle T. The Kauffman bracket polynomial [Formula: see text] of a link L is a regular isotopy invariant of links taking values in the Laurent polynomial ring [Formula: see text]. For each integer i, we associate an integer [Formula: see text], which is obtained from the Kauffman bracket polynomial [Formula: see text] of L. Let d be the greatest common divisor of [Formula: see text] and [Formula: see text] for a 4-tangle T. We will show that there exists a natural number n such that d divides [Formula: see text] if the tangle T is embedded in a link L.

Kauffman bracket versus Jones polynomial skein modules

This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by Melvin and Kirby that underlie the quantum group-based Reshetikhin–Turaev model for [Formula: see text] Chern–Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented.

The action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the 3-twist knot complement

We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern–Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, the first author, and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a detailed example in a part of Chern–Simons theory where examples are scarce.

Non-semisimple 3-manifold invariants derived from the Kauffman bracket

We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of \mathfrak{sl_2} using purely combinatorial methods based on Temperley–Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten–Reshetikhin–Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.

Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

The Kauffman Skein Algebra of the Torus

Abstract We give a presentation of the Kauffman (BMW) skein algebra of the torus. This algebra is the “type $BCD$” analogue of the Homflypt skein algebra of torus, which was computed in earlier work of the 1st and 3rd authors [17]. This suggests the existence of a “type $BCD$” version of the Hall algebra of an elliptic curve [4]. In the appendix, we show this presentation is compatible with the Frohman–Gelca description of the Kauffman bracket (Temperley–Lieb) skein algebra of the torus [12].

A Generalisation of the Kauffman Bracket Polynomial to Determine and Analyse Structural Elements in a RNA Secondary Structure

A Generalisation of the Kauffman Bracket Polynomial to Determine and Analyse Structural Elements in a RNA Secondary Structure

Guts, volume and skein modules of 3-manifolds

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket func

On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra.

Presentations of the Roger–Yang generalized skein algebra

We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmuller space and generalizes the construction of the Kauffman bracket skein algebra. In this paper, we also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory.

A basis for the Kauffman skein module of the product of a surface and a circle

The Kauffman bracket skein module $S(M)$ of a 3-manifold $M$ is a $\mathbb{Q}(A)$-vector space spanned by links in $M$ modulo the so-called Kauffman relations. In this article, for any closed oriented surface $\Sigma$ we provide an explicit spanning family for the skein modules $S(\Sigma\times S^1)$. Combined with earlier work of Gilmer and Masbaum, we answer their question about the dimension of $S(\Sigma\times S^1)$ being $2^{2g+1} + 2g -1$.

The Jones polynomials of three-bridge knots via Chebyshev knots and billiard table diagrams

This work presents formulas for the Kauffman bracket and Jones polynomials of three-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of two-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.

Parity, virtual closure and minimality of knotoids

In this paper, we study parity in planar and spherical knotoids in relation to virtual knots. We introduce a planar version of the parity bracket polynomial for planar knotoids. We show that the virtual closure map (a map from the set of knotoids in [Formula: see text] to the set of virtual knots of genus at most one) is not surjective, by utilizing the surface bracket polynomial of virtual knots. We give specific examples of virtual knots that are not in the image of the virtual closure map. Turaev conjectured that minimal diagrams of knot-type knotoids have zero height. We prove this conjecture by using the results of Nikonov and Manturov induced by parities of virtual knots.

Coefficients of catalan states without nesting and maximal number of returns

We find and study properties of the coefficients [Formula: see text] and [Formula: see text] of Catalan states with the maximal number of returns and no nesting in the Kauffman bracket sum of an [Formula: see text] lattice crossing [Formula: see text]. We show that, up to a power of [Formula: see text], [Formula: see text] and [Formula: see text] are polynomials in variable [Formula: see text] with unimodal and palindromic coefficients. Our results can be viewed as a step toward obtaining closed form formulas for coefficients of an arbitrary Catalan state obtained from [Formula: see text].

Faithfullness of Geometric Action of Skein Algebras

We show that the action of the Kauffman bracket skein algebra of a surface Σ on the skein module of the handlebody bounded by Σ is faithful if and only if the quantum parameter is not a root of 1.

Finite type invariants for knotoids

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.

Dimension and Trace of the Kauffman Bracket Skein Algebra

Let F F be a finite type surface and ζ \zeta a complex root of unity. The Kauffman bracket skein algebra K ζ ( F ) K_\zeta (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 Field Theories. We compute the rank and trace of K ζ ( F ) K_\zeta (F) over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of F F .

SU(3)-skein algebras and webs on surfaces

The \(SU_3\)-skein algebra of a surface F is spanned by isotopy classes of certain framed graphs in \(F\times I\) called 3-webs subject to the skein relations encapsulating relations between \(U_q(sl(3))\)-representations. It is expected that their theory parallels that of the Kauffman bracket skein algebras. We make the first step towards developing that theory by proving that the reduced \(SU_3\)-skein algebra of any surface of finite type is finitely generated. We achieve that result by developing a theory of canonical forms of webs in surfaces. Specifically, we show that for any ideal triangulation of F every reduced 3-web can be uniquely decomposed into unions of pyramid formations of hexagons and disjoint arcs in the faces of the triangulation with possible additional “crossbars” connecting their edges along the ideal triangulation. We show that such canonical position is unique up to “crossbar moves”. That leads us to an associated system of coordinates for webs in triangulated surfaces (counting intersections of the web with the edges of the triangulation and their rotation numbers inside of the faces of the triangulation) which determine a reduced web uniquely. Finally, we relate our skein algebras to \({\mathcal {A}}\)-varieties of Fock–Goncharov and to \(\text {Loc}_{SL(3)}\)-varieties of Goncharov–Shen. We believe that our coordinate system for webs is a manifestation of a (quantum) mirror symmetry conjectured by Goncharov–Shen.