Kauffman Bracket Research Articles

The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal

We give an explicit basis [Formula: see text] of the quotient of the Kauffman bracket skein algebra [Formula: see text] on a surface [Formula: see text] by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus [Formula: see text] into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal.

Pseudo knots and an obstruction to cosmetic crossings

Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct an obstruction to cosmetic crossings in classical links.

The Torelli group and the Kauffman bracket skein module

AbstractWe 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.

Three approaches to a bracket polynomial for singular links

Three approaches to a bracket polynomial for singular links

On Positivity of Kauffman Bracket Skein Algebras of Surfaces

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

Multi-crossing number for knots and the Kauffman bracket polynomial

AbstractA multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3: $$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$ We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.

A skein action of the symmetric group on noncrossing partitions

We introduce and study a new action of the symmetric group $${\mathfrak {S}}_n$$Sn on the vector space spanned by noncrossing partitions of $$\{1, 2,\ldots , n\}$${1,2,ź,n} in which the adjacent transpositions $$(i, i+1) \in {\mathfrak {S}}_n$$(i,i+1)źSn act on noncrossing partitions by means of skein relations. We characterize the isomorphism type of the resulting module and use it to obtain new representation-theoretic proofs of cyclic sieving results due to Reiner---Stanton---White and Pechenik for the action of rotation on various classes of noncrossing partitions and the action of K-promotion on two-row rectangular increasing tableaux. Our skein relations generalize the Kauffman bracket (or Ptolemy relation) and can be used to resolve any set partition as a linear combination of noncrossing partitions in a $${\mathfrak {S}}_n$$Sn-equivariant way.

Presentations of Roger and Yang’s Kauffman bracket arc algebra

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.

Roger and Yang’s Kauffman bracket arc algebra is finitely generated

Generalizing the construction of the Kauffman bracket skein algebra, Roger and Yang defined the Kauffman bracket arc algebra of a punctured surface, so that it is a quantization of the decorated Teichmüller space. We provide an explicit, finite set of generators for their algebra.

Tabulating knots in the thickened Klein bottle

We tabulate all knots in the oriented thickened Klein bottle having diagrams with three crossings and less. For proving that the knots are distinct, we use a generalization of the Kauffman bracket polynomial in four variables.

Frobenius algebras derived from the Kauffman bracket skein algebra

The Kauffman bracket skein algebra of a compact oriented surface when the variable [Formula: see text] in the Kauffman bracket is set equal to [Formula: see text], where [Formula: see text] is an odd counting number, is a central extension of the ring of [Formula: see text]-characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

Double affine Hecke algebras and generalized Jones polynomials

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

Cayley Factorization and the Area Principle

In Sturmfels and Whiteley (J Symb Comput 11(5):439---453, 1991), it is proven that any multihomogenous bracket polynomial with integer coefficients can be interpreted by a ruler construction when introducing simple non-degeneracy conditions. This problem is called (generalized) Cayley factorization. This allows for geometrically interpreting many projective invariant properties. We reprove the above statement in rank 3 giving a better bound on the size of the non-degeneracy conditions. The constant factor in the bound is essentially reduced from 105 to 9. The algorithm described is concise enough to be implemented on a computer. With this algorithm, interpreting a common condition for six points to lie on a common conic is very close to Pascal's construction (see Apel, in: The geometry of brackets and the area principle, Dissertation, TU Munchen, 2014).

Bollobás–Riordan and Relative Tutte Polynomials

We establish a relation between the Bollobas–Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.

Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group of S to SL_2(C), or in a twisted version of this character variety. The proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors. These miraculous cancellations also play a fundamental role in subsequent work of the authors, where novel examples of representations of the skein algebra are constructed.

On the volume conjecture for polyhedra

We formulate a generalization of the volume conjecture for planar graphs. Denoting by $$\langle \Gamma , c \rangle ^{\mathrm {U}}$$ the Kauffman bracket of the graph $$\Gamma $$ whose edges are decorated by real “colors” c, the conjecture states that, under suitable conditions, certain evaluations of $$\langle \Gamma ,\lfloor kc \rfloor \rangle ^{\mathrm {U}}$$ grow exponentially as $$k\rightarrow \infty $$ and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is $$\Gamma $$ (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar graphs, generalizing the Gordon–Schulten recursion for the quantum 6j-symbols. Assuming that $$\langle \Gamma ,\lfloor kc \rfloor \rangle ^{\mathrm {U}}$$ does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schläfli equation); moreover, any small perturbation of the volume function that is still a solution to these equations, is a perturbation by an additive constant. In the appendix we also provide a proof outlined elsewhere of the conjecture for an infinite family of planar graphs including the tetrahedra.

Surface pole bracket polynomials of virtual knots and twisted knots

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 links. This invariant is deeply related to the surface states. In this paper, we introduce the notion of surface pole bracket polynomials for link diagrams in closed surfaces, as a generalization of surface bracket polynomials by Dye and Kauffman. The polynomials induce the invariant of twisted links defined by the author before as a generalization of Dye, Kauffman and Miyazawa's polynomial invariant. Furthermore we discuss a relationship between curves in surface pole states and variables of the polynomial invariant.

On Kauffman bracket skein modules at roots of unity

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.

Uniqueness of reduced alternating rational 3-tangle diagrams

Tangles were introduced by J. Conway. In 1970, he proved that every rational 2-tangle defines a rational number and two rational 2-tangles are isotopic if and only if they have the same rational number. So, from Conway's result we have a perfect classification for rational 2-tangles. However, there is no similar theorem to classify rational 3-tangles. In this paper, we introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.

Chebyshev polynomials and the Frohman–Gelca formula

Using Chebyshev polynomials, C. Frohman and R. Gelca introduced a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones–Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which emphasizes and demystifies the role played by Chebyshev polynomials.