Skein Relations Research Articles

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.

Categories generated by a trivalent vertex

This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate pivotal tensor categories over generated by a symmetric self-dual simple object X and a rotationally invariant morphism $$1 \rightarrow X \otimes X \otimes X$$ . Our main result is that the only trivalent categories with $$\dim {\text {Hom}}(1 \rightarrow X^{\otimes n})$$ bounded by 1, 0, 1, 1, 4, 11, 40 for $$0 \le n \le 6$$ are quantum SO(3), quantum $$G_2$$ , a one-parameter family of free products of certain Temperley-Lieb categories (which we call ABA categories), and the H3 Haagerup fusion category. We also prove similar results where the map $$1 \rightarrow X^{\otimes 3}$$ is not rotationally invariant, and we give a complete classification of nondegenerate braided trivalent categories with dimensions of invariant spaces bounded by the sequence 1, 0, 1, 1, 4. Our main techniques are a new approach to finding skein relations which can be easily automated using Grobner bases, and evaluation algorithms which use the discharging method developed in the proof of the 4-color theorem.

On the formulae for the colored HOMFLY polynomials

We provide methods to compute the colored HOMFLY polynomials of knots and links with symmetric representations based on the linear skein theory. By using diagrammatic calculations, several formulae for the colored HOMFLY polynomials are obtained. As an application, we calculate some examples for hyperbolic knots and links, and we study a generalization of the volume conjecture by means of numerical calculations. In these examples, we observe that asymptotic behaviors of invariants seem to have relations to the volume conjecture.

An Invariant of Colored Links via Skein Relation

In this note, we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links.

Exchange relation planar algebras of small rank

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras: subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.

Refined composite invariants of torus knots via DAHA

We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the skein theory of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries of the DAHA-superpolynomials of torus knots. The latter are conjecturally related to the HOMFLY-PT homology. At the end, we construct two DAHA-hyperpolynomials extending the DAHA-Jones polynomials of type E closely related to the Deligne-Gross approach to the exceptional root systems; this theme is of experimental nature.

HOMFLY Polynomial from a Generalized Yang–Yang Function

Starting from the free field realization of Kac–Moody Lie algebra, we define a generalized Yang–Yang function. Then for the Lie algebra of type $$A_{n}$$ , we derive braiding and fusion matrix by braiding the thimble from the generalized Yang–Yang function. One can construct a knots invariant H(K) from the braiding and fusion matrix. It is an isotropy invariant and obeys a skein relation. From them, we show that the corresponding knots invariant is HOMFLY polynomial.

Chern-Simons Path Integrals in S2 × S1

Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\).

The tail of a quantum spin network

The tail of a sequence \(\{P_n(q)\}_{n \in \mathbb {N}}\) of formal power series in \(\mathbb {Z}[[q]]\) is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of \(P_n\). This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews–Gordon identities for the two-variable Ramanujan theta function as well to corresponding identities for the false theta function.

Compactifications of character varieties and skein relations on conformal blocks

Let $M_C(G)$ be the moduli space of semistable principal $G-$bundles over a smooth curve $C$. We show that a flat degeneration of this space $M_{C_{\Gamma}}(G)$ associated to a singular stable curve $C_{\Gamma}$ contains the free group character variety $\mathcal{X}(F_g, G)$ as a dense, open subset, where $g = genus(C).$ In the case $G = SL_2(\mathbb{C})$ we describe the resulting compactification explicitly, and in turn we conclude that the coordinate ring of $M_{C_{\Gamma}}(SL_2(\mathbb{C}))$ is presented by homogeneous skein relations. Along the way, we prove the parabolic version of these results over stable, marked curves $(C_{\Gamma}, \vec{p}_{\Gamma})$.

On skein relations in class S theories

Loop operators of a class S theory arise from networks on the corresponding Riemann surface, and their operator product expansions are given in terms of the skein relations, that we describe in detail in the case of class S theories of type A. As two applications, we explicitly determine networks corresponding to dyonic loops of $N=4$ $SU(3)$ super Yang-Mills, and compute the superconformal index of a nontrivial network operator of the $T_3$ theory.

Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs

Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula whose terms are parametrized by the perfect matchings of a snake graph. In this paper, we continue our study of snake graphs from a combinatorial point of view. We advance the study of abstract snake graphs by introducing the notions of abstract band graphs, self-crossings of abstract snake graphs as well as their resolutions. We show that there is a bijection between the set of perfect matchings of crossing or self-crossing snake graphs and the set of perfect matchings of the resolution of the crossing. In the situation where the snake and band graphs are coming from arcs and loops in a surface without punctures, we obtain a new proof of skein relations in the corresponding cluster algebra.

On the derivation of the HOMFLYPT polynomial invariant for fluid knots

By using and extending earlier results (Liu & Ricca,J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.

Topologically massive Yang–Mills theory and link invariants

Topologically massive Yang–Mills theory is studied in the framework of geometric quantization. Since this theory has a mass gap proportional to the topological mass m, Yang–Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern–Simons theory with level number k. In this paper, the near Chern–Simons limit is studied where the distance is large enough to give an almost topological theory, with a small contribution from the Yang–Mills term. It is shown that this almost topological theory consists of two copies of Chern–Simons with level number k/2, very similar to the Chern–Simons splitting of topologically massive AdS gravity. Also, gauge invariance of these half-Chern–Simons theories is discussed. As m approaches to infinity, the split parts add up to give the original Chern–Simons term with level k. Reduction of the phase space is discussed in this limit. Finally, a relation between the observables of topologically massive Yang–Mills theory and Chern–Simons theory is shown. One of the two split Chern–Simons pieces is shown to be associated with Wilson loops while the other with 't Hooft loops. This allows one to use skein relations to calculate topologically massive Yang–Mills theory observables in the near Chern–Simons limit.

Defect networks and supersymmetric loop operators

We consider topological defect networks with junctions in A N − 1 Toda CFT and the connection to supersymmetric loop operators in $$ \mathcal{N}=2 $$ theories of class $$ \mathcal{S} $$ on a four-sphere. Correlation functions in the presence of topological defect networks are computed by exploiting the monodromy of conformal blocks, generalising the notion of a Verlinde operator. Concentrating on a class of topological defects in A 2 Toda theory, we find that the Verlinde operators generate an algebra whose structure is determined by a set of generalised skein relations that encode the representation theory of a quantum group. In the second half of the paper, we explore the dictionary between topological defect networks and supersymmetric loop operators in the $$ \mathcal{N}={2}^{*} $$ theory by comparing to exact localisation computations. In this context, the the generalised skein relations are related to the operator product expansion of loop operators.

Polynomials, Binary Trees, and Positive Braids

In knot theory, a common task is to take a given knot diagram and generate from it a polynomial. One method for accomplishing this is to employ a skein relation to convert the knot into a type of labeled binary tree and from this tree derive a two-variable polynomial. The purpose of this paper is to determine, in a simplified setting, which polynomials can be generated from labeled binary trees. We give necessary and sufficient conditions for a polynomial to be constructible in this fashion and we will provide a method for reconstructing the generating tree from such a polynomial. We conclude with an application of this theorem to a class of knots and links given by closed positive braids.

A lattice of finite-type invariants of virtual knots

We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based horizontally upon the Polyak algebra and extended vertically using Manturov's functorial map $f$. For each $n$, the $n$-th vertical line in the lattice contains an infinite dimensional subspace of Kauffman finite-type invariants of degree $n$. Moreover, the lattice contains infinitely many inequivalent extensions of the Conway polynomial to long virtual knots, all of which satisfy the same skein relation. Bounds for the rank of each group in the lattice are obtained.

ALLOWABLE CLOSED SURFACE EVALUATIONS FROM TOPOLOGICAL QUANTUM FIELD THEORY

Every 2D TQFT evaluates closed 2D surfaces to an element of the base field. These evaluations appear as a fundamental part of the skein relations induced by the TQFT, dating back to Bar-Natan. We definitively classify what sets of closed surface evaluations may arise from a 2D TQFT. Our answer reveals a deep relationship between achievable evaluations and the theory of symmetric polynomials. We then apply our results to precisely determine which 2D TQFTs have an associated Frobenius algebra that may be realized as the algebra of all surfaces with boundary S1, modulo surfaces that are "evaluated similarly" by the TQFT.

On the Tutte–Krushkal–Renardy polynomial for cell complexes

Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte–Krushkal–Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality identity as the original polynomial. We find that evaluating the Tutte–Krushkal–Renardy along a certain line gives the Bott polynomial. Finally we prove skein relations for the Tutte–Krushkal–Renardy polynomial.

HOMFLY Polynomial Invariants of Torus Knots and Bosonic (q, p)-Calculus

For the one-parameter Alexander (Jones) skein relation we introduce the Alexander (Jones) “bosonic” q-numbers, and for the two-parameter HOMFLY skein relation we propose the HOMFLY “bosonic” (q, p)-numbers (“bosonic” numbers connected with deformed bosonic oscillators). With the help of these deformed “bosonic” numbers, the corresponding skein relations can be reproduced. Analyzing the introduced “bosonic” numbers, we point out two ways of obtaining the two-parameter HOMFLY skein relation (“bosonic” (q, p)-numbers) from the one-parameter Alexander and Jones skein relations (from the corresponding “bosonic” q-numbers). These two ways of obtaining the HOMFLY skein relation are equivalent.