Arborescent Knots Research Articles

Tangle blocks in the theory of link invariants

The central discovery of 2d conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire huge Hilbert space of states. Somewhat similar, when a link diagram is glued from tangles, the link polynomial is a multilinear combination of tangle blocks summed over just a few representations of intermediate states. This turns to be a powerful approach because the same tangles appear as constituents of very different knots so that they can be extracted from simpler cases and used in more complicated ones. So far this method has been technically developed only in the case of arborescent knots, but, in fact, it is much more general. We begin a systematic study of tangle blocks by detailed consideration of some archetypical examples, which actually lead to non-trivial results, far beyond the reach of other techniques. At the next level, the tangle calculus is about gluing of tangles, and functorial mappings from Hom(tangles). Its main advantage is an explicit realization of multiplicative composition structure, which is partly obscured in traditional knot theory.

Checks of integrality properties in topological strings

Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N ) and SO(N ) adjoint representations [1] are useful to verify Marino’s integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.

Construcción de conocimiento Rizomático en las ciencias de la gestión: Implicancias para la comprensión de organizaciones de alta confiabilidad. Building rhizomic knowledge in organization sciences: implications to understand high reliability organization

De la arborescencia del mainstream positivista a la actual diversidad paradigmática, las ciencias de la gestión se nutren de múltiples fuentes epistemológicas. El presente trabajo pone en evidencia el carácter rizomático de los estudios organizacionales y sus implicancias para comprender las Organizaciones de Alta Confiabilidad (HROs).
 En primer lugar se analizan las fuentes epistemológicas de la disciplina para dar cuenta del pasaje del mainstream positivista hacia la diversidad paradigmática. En la actualidad, la teoría de la organización es un campo disciplinar rizomático, tanto por sus enfoques epistemológicos, la multiplicidad de temas de estudio, la profusión de fuentes disciplinarias como por la diversidad de estrategias metodológicas. Naturaleza epistemológica rizomática que convive con ciertos nudos arborescentes.
 En segundo lugar, el foco de estudio se traslada hacia las HROs, fenómeno organizacional inherente a las sociedades del riesgo que constituye un desafío epistemológico para las teorías del organizar, tanto para comprender la naturaleza de las HROs como para explicar su performance.
 Finalmente, se plantea una serie de desafíos epistemológicos a los que se suman las especificidades del universo organizacional argentino, caracterizado por la profusión de organizaciones riesgosas, la recurrencia de accidentes y la incipiente presencia de equipos de investigación que desarrollan el campo disciplinar.

Tabulating knot polynomials for arborescent knots

Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys. B 899 395–413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials.

On moduli space of symmetric orthogonal matrices and exclusive Racah matrixS¯for representation R = [3,1] with multiplicities

Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of arXiv:1612.00422, we obtain the exclusive Racah matrix $\bar S$ for the currently-front-line case of representation $R=[3,1]$ with non-trivial multiplicities, where it is actually operator valued, i.e. depends on the choice of basises in the intertwiner spaces. Effective field theory for arborescent knots in this case possesses gauge invariance, which is not yet properly described and understood. Because of this lack of knowledge a big part (about a half) of $\bar S$ needs to be reconstructed from orthogonality conditions. Therefore we discuss the abundance of symmetric orthogonal matrices, to which $\bar S$ belong, and explain that dimension of their moduli space is also about a half of that for the ordinary orthogonal matrices. Thus the knowledge approximately matches the freedom and this explains why the method can work -- with some limited addition of educated guesses. A similar calculation for $R=[r,1]$ for $r>3$ should also be doable.

Factorization of differential expansion for antiparallel double-braid knots

Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations $R=[r^s]$ we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of $R=[33]$. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices $\bar S$ and $S$ -- what allows to calculate $[33]$-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations described in detail are only the contributions of the single- and two-floor pyramids, the way to proceed is explicitly illustrated by the examples of $R=[44]$ and $R=[55]$. This solves the difficult part of the problem, but the last tedious step towards explicit formulas for generic exclusive rectangular Racah matrices still remains to be made.

Quantum Racah matrices and 3-strand braids in irreps R with |R| = 4

We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R=[2,2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6x6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate eigenvalues and is evaluated by the highest weight method. Together with the much harder calculation for R=[3,1] in arXiv:1605.02313 and with the new method to extract exclusive matrices S and \bar S from the inclusive ones, this completes the story of Racah matrices for |R|\leq 4 and allows one to calculate and investigate the corresponding colored HOMFLY polynomials for arbitrary 3-strand and arborescent knots.

Racah matrices and hidden integrability in evolution of knots

We construct a general procedure to extract the exclusive Racah matrices S and S¯ from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R=[1], [2], [3] and [2,2]. The matrices S and S¯ relate respectively the maps (R⊗R)⊗R¯⟶R with R⊗(R⊗R¯)⟶R and (R⊗R¯)⊗R⟶R with R⊗(R¯⊗R)⟶R. They are building blocks for the colored HOMFLY polynomials of arbitrary arborescent (double fat) knots. Remarkably, the calculation realizes an unexpected integrability property underlying the evolution matrices.

Universal Racah matrices and adjoint knot polynomials: Arborescent knots

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso–Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SUN) and Kauffman (SON) polynomials. For E8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the “eigenvalue conjecture”, which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6×6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.

Small Seifert fibered surgery on hyperbolic pretzel knots

The study of exceptional surgery on hyperbolic knots has been well developed over the last quarter century. One particularly well studied problem is that of exceptional surgery on arborescent knots, which include Montesinos knots and pretzel knots. Thanks to the positive solution to the geometrization conjecture by Perelman [38; 39; 40], any exceptional surgery is either reducible, toroidal, or a small Seifert fibered space. Exceptional surgeries on hyperbolic arborescent knots of length 4 or greater have been classified by Wu [52], as have exceptional surgeries on hyperbolic 2–bridge knots; see Brittenham and Wu [7]. It has been shown that no hyperbolic arborescent knot admits a reducible surgery (see Wu [48]), and toroidal surgeries on hyperbolic arborescent knots of length three are completely classified; see Wu [51].

ON NON-ISOTOPIC SPANNING SURFACES FOR A CLASS OF ARBORESCENT KNOTS

We use the methods of Hedden, Juhász and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots Kn which will bound at least 22n-1 non-isotopic minimal spanning surfaces of genus n.

Exceptional Dehn surgery on large arborescent knots

A Dehn surgery on a knot K in S is exceptional if it produces a reducible, toroidal or Seifert fibred manifold. It is known that a large arborescent knot admits no such surgery unless it is a type II arborescent knot. The main theorem of this paper shows that up to isotopy there are exactly three large arborescent knots admitting exceptional surgery, each of which admits exactly one exceptional surgery, producing a toroidal manifold.

Hopf plumbing and minimal diagrams

We determine minimal diagrams and braid indices for fibered arborescent knots arising from plumbing positive or negative Hopf bands along trees.

THE BRAID INDEX AND THE GROWTH OF VASSILIEV INVARIANTS

We use the new approach of braiding sequences to prove exponential upper bounds for the number of Vassiliev invariants on knots with bounded braid index, bounded bridge number and arborescent knots. We prove, that any Vassiliev invariant of degree k is determined by its values on knots with braid index at most k + 1.

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Dehn surgery on arborescent knots and links—A survey

In this survey, we present some recent results about Dehn surgeries on arborescent knots and links. Arborescent links are also known as algebraic links [1, 2] and star links [3]. The set of arborescent knots and links is a large class, including all 2-bridge links and Montesinos links. They have been studied by many people, see e.g. [4,2,5,3,6, 7]. We will give some definitions below. The reader is referred to [8, 9] for more detailed background material for 3-manifold topology, to [2,1,4,10] for arborescent tangles and links, to [11] for hyperbolic manifolds, and to [12] for essential laminations and branched surfaces.

SURGERY AND FOLIATIONS OF KNOT COMPLEMENTS

An interesting class of knots have complement with a remarkably simple topological description. This class includes all the arborescent knots with only even weights hence, in particular, the two bridge knots and many knots of ten or fewer crossings. For these knots, there are choices of minimal genus Seifert surfaces S such that all taut, depth one foliations of the knot complement, having S as sole compact leaf, can be classified up to isotopy. These foliations correspond exactly to the lattice points over the open faces of the unit ball in a Thurston-like norm on the relative homology of the complement of S.