Tangle Diagrams Research Articles

Tangles in affine Hecke algebras

Framed oriented [Formula: see text]-tangle diagrams in the annulus, subject to the Homfly skein relations, are used to produce an algebra isomorphic to the affine Hecke algebras [Formula: see text] of type [Formula: see text]. The use of closed curves and braids gives neat pictures for central elements in the algebras.

Over then under tangles

Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful “gliding ideas” in the literature — sometimes in disguise — such as the Drinfel’d double construction, Enriquez’s work on quantization of Lie bialgebras, and Audoux and Meilhan’s classification of welded homotopy links.

Chern–Simons theory, link invariants and the Askey–Wilson algebra

The occurrence of the Askey–Wilson (AW) algebra in the SU(2) Chern–Simons (CS) theory and in the Reshetikhin–Turaev (RT) link invariant construction with quantum algebra Uq(su2) is explored. Tangle diagrams with three strands with some of them enclosed in a spin-1/2 closed loop are associated to the generators of the AW algebra. It is shown in both the CS theory and RT construction that the link invariant of these tangles obey the relations of the AW generators. It follows that the expectation values of certain Wilson loops in the CS theory satisfy relations dictated by the AW algebra and that the link invariants do not distinguish the corresponding linear combinations of links.

The colored Jones polynomials as vortex partition functions

We construct 3D mathcal{N} = 2 abelian gauge theories on mathbbm{S} 2 × mathbbm{S} 1 labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in mathbbm{S} 3. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on mathbbm{S} 3, and then our construction provides an explicit correspondence between 3D mathcal{N} = 2 abelian gauge theories and 3D SU(2) Chern-Simons gauge theories. We verify, in particular, the applicability of our constructions to a class of tangle diagrams of 2-bridge knots with certain specific twists.

A cobordism realizing crossing change on [formula omitted] tangle homology and a categorified Vassiliev skein relation

We discuss degree-preserving crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we investigate a sum of cobordisms that yields a morphism between complexes of two diagrams related by a change of crossing, which we call the “genus-one morphism.” We prove that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain invariants for singular tangles that extend sl(2) tangle homologies; examples include Lee homology, Bar-Natan homology, and Naot's universal Khovanov homology as well as Khovanov homology with arbitrary coefficients. We also verify that the invariant satisfies categorified analogues of Vassiliev skein relation and the FI relation.

Generating links that are both quasi-alternating and almost alternating

We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all non-alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (6), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (1), 2015) is false.

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology.

Random walks on the BMW monoid: an algebraic approach

We consider Metropolis-based systematic scan algorithms for generating Birman–Murakami–Wenzl (BMW) monoid basis elements of the BMW algebra. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We also consider the Brauer algebra and use Metropolis-based scans to generate Brauer diagrams, giving rise to random walks on perfect matchings. Taking an algebraic perspective, we translate these walks into left multiplication operators in the BMW algebra and so give an algebraic interpretation of the Metropolis algorithm in this setting.

Complexities of a knitting pattern

A knitting pattern is a tangle diagram in the square and a knitting is a tessellation of a knitting pattern in the plane. In this paper, a classification of the knitting patterns is done in terms of virtual links and some complexities of a knitting pattern by some topological invariants are studied. Generalizations to a chain pattern and a graph-tangle pattern are also discussed.

On the classification of a large set of rational 3-tangle diagrams

We note that a rational [Formula: see text]-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational [Formula: see text]-tangle diagrams up to isotopy: [B. Kwon, An algorithm to classify rational [Formula: see text]-tangles, J. Knot Theory Ramifications 24(1) (2015) 1550004]. However, there is no perfect classification about rational [Formula: see text]-tangle diagrams such as the classification of rational [Formula: see text]-tangle diagrams corresponding to rational numbers. In this paper, we classify a large set of rational [Formula: see text]-tangles which are generated by only three generators.

2-Tangle replacements and adequate diagrams

We consider [Formula: see text]-tangle replacements of link diagrams, i.e. replacing a crossing of a link diagram [Formula: see text] with a [Formula: see text]-tangle diagram [Formula: see text]. We show that if [Formula: see text] and [Formula: see text] are adequate then the [Formula: see text]-tangle replacement, denoted by [Formula: see text], is also adequate. This gives a way to obtain infinitely many new minimal crossing diagrams.

On unknotting operations of rotation type

An unknotting operation is a local move on a knot diagram such that any knot diagram can be transformed into a diagram of the unknot by a finite sequence of the operations and Reidemeister moves. In this paper, we introduce a new local move H(T) on a knot diagram which is obtained by the rotation of a tangle diagram T and study their properties. As an application, we prove that the H(T)-move is an unknotting operation for any descending tangle diagram T.

Gamma-polynomial and its generalization to a 2-string tangle polynomial

The zeroth coefficient polynomial of the skein (HOMFLYPT) knot polynomial called the Γ-polynomial is studied from a viewpoint of regular homotopy of knot diagrams. In particular, an elementary existence proof of the knot invariance of the Γ-polynomial is given. After observing that there are three types for 2-string tangle diagrams, the Γ-polynomial is generalized to a polynomial invariant of a 2-string tangle. As an application, we have a new proof of the assertion that Kinoshita's θ-curve is not equivalent to the trivial θ-curve.

Graphic Lambda Calculus

Graphic lambda calculus, a visual language that can be used for representing untyped lambda calculus, is introduced and studied. It can also be used for computations in emergent algebras or for representing Reidemeister moves of locally planar tangle diagrams. Graphic lambda calculus consists of a class of graphs endowed with moves between them. It might be considered a visual language in the sense of Erwig [1]. The name comes from the fact that it can be used for representing terms and reductions from untyped lambda calculus. Its main move is called the graphic beta move for its relation to the beta reduction in lambda calculus. However, the graphic beta move can be applied outside the “sector” of untyped lambda calculus, and the graphic lambda calculus can be used for other purposes than that of visually representing lambda calculus. For other visual, diagrammatic representations of lambda calculus see the VEX language [2], or Keenan’s website [3]. The motivation for introducing graphic lambda calculus comes from the study of emergent algebras. In fact, my goal is to eventually build a logic system that can be used for the formalization of certain “computations” in emergent algebras. The system can then be applied for a discrete differential calculus that exists for metric spaces with dilations, comprising Riemannian manifolds and sub-Riemannian spaces with very low regularity. Emergent algebras are a generalization of quandles; namely, an emergent algebra is a family of idempotent right quasigroups indexed by the elements of an Abelian group, while quandles are self-distributive idempotent right quasigroups. Tangle diagrams decorated by quandles or racks are a well-known tool in knot theory [4, 5]. In Kauffman [6] knot diagrams are used for representing combinatory logic, thus forming a graphical notation for untyped lambda cal

ON THE PRESENTATION OF A LINK AS A SUM OF TWO DESCENDING TANGLE DIAGRAMS

In this paper, we define the notion of descending tangle diagrams and give a presentation of a link as a sum of two descending tangle diagrams. We also introduce a new invariant of links and find its basic properties.

Ribbon Hopf algebras from group character rings

We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group in the inductive limit, realized as the ring of symmetric functions on countably many variables . Isomorphic as spaces are the character rings Char-O and Char-Sp of the classical matrix subgroups of , the orthogonal and symplectic groups. We also analyse the formal character rings Char-H of algebraic subgroups of , comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type , which have been introduced earlier (Fauser et al.) (these include the orthogonal and symplectic groups as special cases). The set of tangle diagrams encoding manipulations of the group and subgroup characters has many elements deriving from products, coproducts, units and counits as well as different types of branching operators. From these elements we assemble for each a crossing tangle which satisfies the braid relation and which is nontrivial, in spite of the commutative and co-commutative setting. We identify structural elements and verify the axioms to establish that each Char-H ring is a ribbon Hopf algebra. The corresponding knot invariant operators are rather weak, giving merely a measure of the writhe.

Computing with Space: A Tangle Formalism for Chora and Difference

What is computing,simulation, or understanding? Converging from several sources, this seems to be something more primitive than what is meant nowadays by computation, something that was along with us since antiquity (the word choros, chora, denotes space or place and is seemingly the most mysterious notion from Plato, described in Timaeus 48e - 53c) which has to do with cybernetics and with the understanding of the front end visual system. It may have some unexpected applications, also. Here, inspired by Bateson (see Supplementary Material), I explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using a formalism of tangle diagrams.

Momenta of a vortex tangle by structural complexity analysis

A geometric method based on information from structural complexity is presented to calculate linear and angular momenta of a tangle of vortex filaments in Euler flows. For thin filaments under the so-called localized induction approximation the components of linear momentum admit interpretation in terms of projected area. By computing the signed areas of the projected graph diagrams associated with the vortex tangle, we show how to calculate the two momenta of the system by complexity analysis of tangle diagrams. This method represents a novel technique to extract dynamical information of complex systems from geometric and topological properties and provides a potentially useful tool to test the accuracy of numerical methods and investigate scale distribution of fluid dynamical properties of vortex flows.

CALCULATION OF DIHEDRAL QUANDLE COCYCLE INVARIANTS OF TWIST SPUN 2-BRIDGE KNOTS

Carter, Jelsovsky, Kamada, Langford and Saito introduced the quandle cocycle invariants of 2-knots, and calculated the cocycle invariant of a 2-twist-spun trefoil knot associated with a 3-cocycle of the dihedral quandle of order 3. Asami and Satoh calculated the cocycle invariants of twist-spun torus knots τrT(m,n) associated with 3-cocycles of some dihedral quandles. They used tangle diagrams of the torus knots. In this paper, we calculate the cocycle invariants of twist-spun 2-bridge knots τrS(α,β) by a similar method.

Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

The Hecke algebra Hn can be described as the skein R n of (n, n)-tangle diagrams with respect to the framed Homfly relations. This algebra R n contains well-known idempotents Eλ which are indexed by Young diagrams λ with n cells. The geometric closures Qλ of the Eλ in the skein of the annulus are known to satisfy a combinatorial multiplication rule which is identical to the Littlewood–Richardson rule for Schur functions in the ring of symmetric functions. This fact is derived from two skein theoretic lemmas by using elementary determinantal arguments. Previously known proofs depended on results for quantum groups.