Link Polynomials Research Articles

A simple algorithm to compute link polynomials defined by using skein relations

Abstract We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete link invariant are obtained.

Orthogonal Periodic Sequences for the Identification of Functional Link Polynomial Filters

The paper introduces a novel family of deterministic periodic signals, the orthogonal periodic sequences (OPSs), that allow the perfect identification on a finite period of any functional link polynomials (FLiP) filter with the cross-correlation method. The class of FLiP filters is very broad and includes many popular nonlinear filters, as the well-known Volterra and the Wiener nonlinear filters. The novel sequences share many properties of the perfect periodic sequences (PPSs). As the PPSs, they allow the perfect identification of FLiP filters with the cross-correlation method. But, while PPSs exist only for orthogonal FLiP filters, the OPSs allow also the identification of non-orthogonal FLiP filters, as the Volterra filters. In OPSs, the modeled system input can be any persistently exciting sequence and can also be a quantized sequence. Moreover, OPSs can often identify FLiP filters with a sequence period and a computational complexity much smaller that PPSs. The provided experimental results, involving the identification of real devices and of a benchmark model, highlight the potentialities of the proposed OPSs in modeling unknown nonlinear systems.

Oriented local moves and divisibility of the Jones–Kauffman polynomial

For any virtual link [Formula: see text] that may be decomposed into a pair of oriented [Formula: see text]-tangles [Formula: see text] and [Formula: see text], an oriented local move of type [Formula: see text] is a replacement of [Formula: see text] with the [Formula: see text]-tangle [Formula: see text] in a way that preserves the orientation of [Formula: see text]. After developing a general decomposition for the Jones polynomial of the virtual link [Formula: see text] in terms of various (modified) closures of [Formula: see text], we analyze the Jones polynomials of virtual links [Formula: see text] that differ via a local move of type [Formula: see text]. Succinct divisibility conditions on [Formula: see text] are derived for broad classes of local moves that include the [Formula: see text]-move and the double-[Formula: see text]-move as special cases. As a consequence of our divisibility result for the double-[Formula: see text]-move, we introduce a necessary condition for any pair of classical knots to be [Formula: see text]-equivalent.

Stable Alexander polynomials of arborescent links

We study the zeros of Alexander polynomials of three classes of arborescent links. In the first class, the zeros are real (and negative) or modulus one. In the second class, the zeros are real (and positive). In the third class, the zeros are real or modulus one. For this purpose, we modify their Alexander polynomials into other real polynomials, with only real zeros, and use the property that two such polynomials have interlacing real zeros.

Multi-colored Links From 3-Strand Braids Carrying Arbitrary Symmetric Representations

Obtaining HOMFLY-PT polynomials $$H_{R_1,\ldots ,R_l}$$ for arbitrary links with l components colored by arbitrary SU(N) representations $$R_1,\ldots ,R_l$$ is a very complicated problem. For a class of rank r symmetric representations, the [r]-colored HOMFLY-PT polynomial $$H_{[r_1],\ldots ,[r_l]}$$ evaluation becomes simpler, but the general answer lies far beyond our current capabilities. To simplify the situation even more, one can consider links that can be realized as a 3-strand closed braid. Recently (Itoyama et al. in Int J Mod Phys A28:1340009, 2013. arXiv:1209.6304 ), it was shown that $$H_{[r]}$$ for knots realized by 3-strand braids can be constructed using the quantum Racah coefficients (6j-symbols) of $$U_q(sl_2)$$ , which makes easy not only to evaluate such invariants, but also to construct analytical formulas for $$H_{[r]}$$ of various families of 3-strand knots. In this paper, we generalize this approach to links whose components carry arbitrary symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials $$H_{[r_1],[r_2]}$$ for the two-component link L7a3 whose components carry $$[r_1]$$ and $$[r_2]$$ colors. Using our results for exclusive Racah matrices, it is possible to calculate symmetric-colored HOMFLY-PT polynomials of links for the so-called one-looped links, which are obtained from arborescent links by adding a loop. This is a huge class of links that contains the entire Rolfsen table, all 3-strand links, all arborescent links, and, for example, all mutant knots with 11 intersections.

A note on Alexander polynomials of 2-bridge links

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-component, 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice.

The coefficients of the HOMFLY polynomials of links

We study the expressions of p1−c(L)(l), the coefficient of the lowest power of m of the HOMFLY polynomial of a link L with c(L) components, give what these coefficients are if they include exactly two terms for knots and the properties of p1−c(L)(1) for knots as well as properties of higher derivatives of p1−c(L) evaluate at i. We show that there are infinitely many different links (knots), especially infinitely many different pretzel links (knots) to have such coefficients and infinitely many different pretzel links (knots) to have same p1−c(L)(l).

Hyperbolicity of the canonical genus two knots

We classify the non-hyperbolic knots (i.e., torus, satellite and composite knots) of canonical genus at most 2. We use cut-and-paste and combinatorial arguments, but most substantially we develop an approach for (computationally) testing knot primeness using link polynomials.

Functional polynomial multiple-index model

Functional polynomial multiple-index model which includes the functional linear model, functional quadratic model and functional single index model as the special cases is considered. In this paper we propose an iterative technique to estimate the link function and polynomial coefficients nonparametrically. Simulations show that the proposed methods have good prediction performances. The proposed model is also illustrated with tecator data.

Unoriented knot polynomials of torus links as Fibonacci-type polynomials

The focus of this paper is to study the two-variable Kauffman polynomials [Formula: see text] and [Formula: see text], and the one-variable BLM/Ho polynomial [Formula: see text] of [Formula: see text]-torus link as the Fibonacci-type polynomials and to express the Kauffman polynomials in terms of the BLM/Ho polynomial. For this purpose, we prove that each of the examined polynomials of [Formula: see text]-torus link can be determined by a third-order recurrence relation and give the recursive properties of them. We correlate these polynomials with the Fibonacci-type polynomials. By using the relations between the BLM/Ho polynomials and Fibonacci-type polynomials, we express the Kauffman polynomials in terms of the BLM/Ho polynomials.

Transcendental linking polynomials of virtual string links

In this paper, we define transcendental polynomial invariants for two-component virtual string links. One of these invariants is a strictly refinement of the linking polynomial in [M. Xu and H. Gao, Linking polynomials of virtual string links, Sci. China Math. 61(7) (2018) 1287–1302]. It is a homotopy invariant and can distinguish some virtual string links from their mirror images. We also define a transcendental polynomial invariant for two-component flat virtual string links. These invariants can be used to study the periodicity and linking crossing number of virtual string links.

Rational Links and DT Invariants of Quivers

Abstract We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson–Thomas invariants of appropriate quivers that we naturally associate with these links. This shows that the conjectural links–quivers correspondence of Kucharski–Reineke–Stošić–Sułkowski as well as the LMOV conjecture holds for all rational links. Along the way, we extend the links–quivers correspondence to tangles and, thus, explore elements of a skein theory for motivic Donaldson–Thomas invariants.

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.

Q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links

We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with one-column (or one-row) Young diagrams.

Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants

We introduce the generalized equidistant Chebyshev polynomials T(k,h) of kind k of hyperkind h, where k, h are positive integers. They are obtained by a generalization of standard and monic Chebyshev polynomials of the first and second kinds. This generalization is fulfilled in two directions. The horizontal generalization is made by introducing hyperkind ℎ and expanding it to infinity. The vertical generalization proposes expanding kind k to infinity with the help of the method of equidistant coefficients. Some connections of these polynomials with the Alexander knot and link polynomial invariants are investigated.

Quantum Racah matrices up to level 3 and multicolored link invariants

This paper is a next step in the project of systematic description of colored knot and link invariants started in Mironov and Morozov (2015) and Mironov et al. (2017). In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels R 1 ⊗ R 2 ⊗ R 3 ⟶ Q with all possible Q , for | R | ≤ 3 . The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org . Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.

Hoste’s Conjecture and Roots of Link Polynomials

In relation to a conjecture of Hoste on the roots of the Alexander polynomial of alternating knots, we prove that any root z of the Alexander polynomial of a 2-bridge (rational) knot or link satisfies $${| z^{1/2}-z^{-1/2} | < 2}$$ . We extend our result to properties of zeros for some Montesinos knots, and to an analogous statement about the skein polynomial. A similar estimate is derived for alternating 3-braid links.

Differential expansion for link polynomials

The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the 6j-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered.

On the Hilbert polynomial of the HOMFLYPT homology

We prove that the degree of the Hilbert polynomial of the HOMFLYPT homology of a closed braid [Formula: see text] is [Formula: see text], where [Formula: see text] is the number of components of [Formula: see text]. This controls the growth of the HOMFLYPT homology with respect to its polynomial grading. The Hilbert polynomial also reveals a link polynomial hidden in the HOMFLYPT polynomial.

A new approach to Lickorish–Millett type formulae

In this paper, we introduce a new method to prove the Lickorish-Millett type formulae for colored HOMFLY-PT polynomials of links.