Kauffman Bracket Research Articles

An Analog of the Kauffman Bracket Polynomial for Knots in the Non-orientable Thickening of a Non-orientable Surface

An Analog of the Kauffman Bracket Polynomial for Knots in the Non-orientable Thickening of a Non-orientable Surface

Planar decomposition of the HOMFLY polynomial for bipartite knots and links

AbstractThe theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from $$\mathfrak {sl}(2)$$ sl ( 2 ) to $$\mathfrak {sl}(N)$$ sl ( N ) at arbitrary N – but for a special class of bipartite diagrams made entirely from the antiparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters – and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov–Rozansky generalizations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

Generalised knotoids

Abstract In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.

A bracket polynomial for virtual links

In this paper, we describe a method of making an invariant of virtual links, generalizing Jones–Kauffman polynomial of virtual links.

On positivity of Roger–Yang skein algebras

We generalize the positivity conjecture on (Kauffman bracket) skein algebras to Roger–Yang skein algebras. To generalize it, we use Chebyshev polynomials of the first kind to give candidates of positive bases. Moreover, the polynomials form a lower bound in the sense of [9] and [10]. We also discuss a relation between the polynomials and the centers of Roger–Yang skein algebras when the quantum parameter is a complex root of unity.

The quantum Frobenius for character varieties and multiplicative quiver varieties

We prove that quantized multiplicative quiver varieties, quantum character varieties, and Kauffman bracket skein algebras each define sheaves of Azumaya algebras over the smooth loci of their corresponding classical moduli spaces. In the case of skein algebras this establishes a strong form of the Unicity Conjecture of Bonahon and Wong. Our proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as well as a natural nondegeneracy condition satisfied by each of the classical Hamiltonian spaces. We therefore introduce the concepts of Frobenius quantum moment maps, Frobenius Poisson orders and their Hamiltonian reductions, and apply them to the study of Azumaya loci.

Elementary computation of the Jones polynomials for Torus links modulo primes

We present simple congruences modulo [Formula: see text] of the Jones polynomial for the torus link [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text] is a nonzero integer. These congruences are expressed in terms of Chebyshev polynomials and quantum numbers. The approach makes elementary use of the Kauffman bracket and does not require any representation theory.

Tangle equations, the Jones conjecture, slopes of surfaces in tangle complements, and q-deformed rationals

AbstractWe study systems of two-tangle equations $$ \begin{align*}\begin{cases} N(X+T_1)=L_1,\\ N(X+T_2)=L_2, \end{cases}\end{align*} $$ which play an important role in the analysis of enzyme actions on DNA strands.We show that every system of framed tangle equations has at most one-framed rational solution. Furthermore, we show that the Jones unknot conjecture implies that if a system of tangle equations has a rational solution, then that solution is unique among all two-tangles. This result potentially opens a door to a purely topological disproof of the Jones unknot conjecture.We introduce the notion of the Kauffman bracket ratio $\{T\}_q\in \mathbb Q(q)$ of any two-tangle T and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$ . We prove that conjecture for algebraic T. We also prove that for rational T, the brackets $\{T\}_q$ coincide with the q-rationals of Morier-Genoud and Ovsienko.Additionally, we relate systems of tangle equations to the cosmetic surgery conjecture and the nugatory crossing conjecture.

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

Triangular decomposition of $\mathrm{SL}_3$ skein algebras

We give an \mathrm{SL}_3 analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Lê. To any punctured bordered surface, we associate an \mathrm{SL}_3 stated skein algebra which contains the \mathrm{SL}_3 skein algebra of closed webs. These algebras admit natural algebra morphisms associated to the splitting of surfaces along ideal arcs. We give an explicit basis for the \mathrm{SL}_3 stated skein algebra and show that the splitting morphisms are injective and describe their images. By splitting a surface along the edges of an ideal triangulation, we see that the \mathrm{SL}_3 stated skein algebra of any ideal triangulable surface embeds into a tensor product of stated skein algebras of triangles. As applications, we prove that the stated skein algebras do not have zero divisors, we construct Frobenius maps at roots of unity, and we obtain a new proof that Kuperberg's web relations generate the kernel of the Reshetikhin–Turaev functor.

Generalizing the relation between the Kauffman bracket and Jones polynomial

We generalize Kauffman’s famous formula defining the Jones polynomial of an oriented link in [Formula: see text]-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology 26(3) (1987) 395–407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented [Formula: see text]-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki’s [Formula: see text]-analog of the first homology group of a [Formula: see text]-manifold [J. Przytycki, A [Formula: see text]-analogue of the first homology group of a [Formula: see text]-manifold, in Contemporary Mathematics, Vol. 214 (American Mathematical Society, 1998), pp. 135–144]. In certain cases, e.g., for links in submanifolds of rational homology [Formula: see text]-spheres, we will be able to define an epimorphism from the Jones module onto the Kauffman bracket module. For the general case we define suitably graded quotients of the bracket module, which are graded by homology. The kernels define new skein modules measuring the difference between Jones and bracket skein modules. We also discuss gluing in this setting.

A quantum invariant of links in T2× I with volume conjecture behavior

We define a polynomial invariant $J_n^T$ of links in the thickened torus. We call $J^T_n$ the $n$th toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, $J_n^T$ exhibits volume conjecture behavior. We prove the volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions of $J_n^T$, one as a generalized operator invariant we call a pseudo-operator invariant, and another using the Kauffman bracket skein module of the torus. Finally, we show $J^T_n$ produces invariants of biperiodic and virtual links. To our knowledge, $J^T_n$ gives the first example of volume conjecture behavior in a virtual (non-classical) link.

Invariants of Multi-linkoids

In this paper, we extend the definition of a knotoid to multi-linkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the T-invariant in relation with generalized Theta -graphs.

The Kauffman bracket skein module of the complement of (2,2p + 1)-torus knots via braids

In this paper we compute the Kauffman bracket skein module of the complement of (2,2p+1)-torus knots, KBSM(T(2,2p+1)c), via braids. We start by considering geometric mixed braids in S3, the closure of which are mixed links in S3 that represent links in the complement of (2,2p+1)-torus knots, T(2,2p+1)c. Using the technique of parting and combing geometric mixed braids, we obtain algebraic mixed braids, that is, mixed braids that belong to the mixed braid group B2,n and that are followed by their “coset” part, that represents T(2,2p+1)c. In that way we show that links in T(2,2p+1)c may be pushed to the genus 2 handlebody, H2, and we establish a relation between KBSM(T(2,2p+1)c) and KBSM(H2). In particular, we show that in order to compute KBSM(T(2,2p+1)c) it suffices to consider a basis of KBSM(H2) and study the effect of combing on elements in this basis. We consider the standard basis of KBSM(H2) and we show how to treat its elements in KBSM(T(2,2p+1)c), passing through many different spanning sets for KBSM(T(2,2p+1)c). These spanning sets form the intermediate steps in order to reach at the set BT(2,2p+1)c, which, using an ordering relation and the notion of total winding, we prove that it forms a basis for KBSM(T(2,2p+1)c). Note that elements in BT(2,2p+1)c have no crossings on the level of braids, and in that sense, BT(2,2p+1)c forms a more natural basis of KBSM(T(2,2p+1)c) in our setting. We finally consider c.c.o. 3-manifolds M obtained from S3 by surgery along the trefoil knot and we discuss steps needed in order to compute the Kauffman bracket skein module of M. We first demonstrate the process described before for computing the Kauffman bracket skein module of the complement of the trefoil, KBSM(Trc), and we study the effect of braid band moves on elements in the basis of KBSM(Trc). These moves reflect isotopy in M and are similar to the second Kirby moves.The “braid” method that we propose for computing Kauffman bracket skein modules seem promising in computing KBSM of arbitrary c.c.o. 3-manifolds M. The only difficulty lies in finding the sufficient relations that reduce elements in the basis of our underlying genus g-handlebody, Hg. These relations come from combing in the case of knot complements and from combing and braid band moves for 3-manifolds obtained by surgery along a knot in S3. Our aim is to set the necessary background of this “braid” approach in order to compute Kauffman bracket skein modules of arbitrary 3-manifolds.

The homological arrow polynomial for virtual links

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper, we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface’s skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled “whiskers” that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study [Formula: see text]-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu’s characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman–Murasugi–Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an “h-reduced” diagram [Formula: see text] is [Formula: see text].

Homological polynomial coefficients and the twist number of alternating surface links

For $D$ a reduced alternating surface link diagram, we bound the twist number of $D$ in terms of the coefficients of a polynomial invariant. To this end, we introduce a generalization of the homological Kauffman bracket defined by Krushkal. Combined with work of Futer, Kalfagianni, and Purcell, this yields a bound for the hyperbolic volume of a class of alternating surface links in terms of these coefficients.

The Kauffman bracket skein module of the lens spaces via unoriented braids

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces [Formula: see text], KBSM([Formula: see text]), for [Formula: see text]. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, [Formula: see text], which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, [Formula: see text], for knots and links in ST, via a unique Markov trace constructed on [Formula: see text]. The universal invariant [Formula: see text] is equivalent to the KBSM(ST). For passing now to the KBSM([Formula: see text]), we impose on [Formula: see text] relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in [Formula: see text] but not in ST, and which reflect the surgery description of [Formula: see text], obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM([Formula: see text]). We first present the solution for the case [Formula: see text], which corresponds to obtaining a new basis, [Formula: see text], for KBSM([Formula: see text]) with [Formula: see text] elements. We note that the basis [Formula: see text] is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case [Formula: see text], we first show how the new basis [Formula: see text] of KBSM([Formula: see text]) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case [Formula: see text]. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.

Strong Positivity for the Skein Algebras of the 4-Punctured Sphere and of the 1-Punctured Torus

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the $${\textit{SL}}_2$$ character variety of a topological surface. We realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov–Witten theory and applied to a complex cubic surface. Using this result, we prove the positivity of the structure constants of the bracelets basis for the skein algebras of the 4-punctured sphere and of the 1-punctured torus. This connection between topology of the 4-punctured sphere and enumerative geometry of curves in cubic surfaces is a mathematical manifestation of the existence of dual descriptions in string/M-theory for the $$\mathcal {N}=2$$ $$N_f=4$$ SU(2) gauge theory.

The Jones polynomial from a Goeritz matrix

AbstractWe give an explicit algorithm for calculating the Kauffman bracket of a link diagram from a Goeritz matrix for that link. Further, we show how the Jones polynomial can be recovered from a Goeritz matrix when the corresponding checkerboard surface is orientable, or when more information is known about its Gordon–Litherland form. In the process we develop a theory of Goeritz matrices for cographic matroids, which extends the bracket polynomial to any symmetric integer matrix. We place this work in the context of links in thickened surfaces.

On invariants for surface-links in entropic magmas via marked graph diagrams

Niebrzydowski and Przytycki defined a Kauffman bracket magma and constructed the invariant [Formula: see text] of framed links in [Formula: see text]-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is obtained from the Kauffman bracket polynomial by the multiplication of indeterminate and it is an ambient isotopy invariant for links. In this paper, we reformulate the multiplication by using a map from the set of framed links to a Kauffman bracket magma in order that [Formula: see text] is invariant for links in [Formula: see text]-space. We define a generalization of a Kauffman bracket magma, which is called a marked Kauffman bracket magma. We find the conditions to be invariant under Yoshikawa moves except the first one and use a map from the set of admissible marked graph diagrams to a marked Kauffman bracket magma to obtain the invariant for surface-links in [Formula: see text]-space.