HOMFLY Research Articles

A relation between Milnor’s μ-invariants and HOMFLYPT polynomials

Polyak showed that any Milnor’s [Formula: see text]-invariant of length 3 can be represented as a combination of the Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called [Formula: see text]-invariants. We show that any Milnor’s [Formula: see text]-invariant of length [Formula: see text] can be represented as a combination of the HOMFLYPT polynomials of knots obtained from the string link by some operation, if all [Formula: see text]-invariants of length [Formula: see text] vanish. Moreover, [Formula: see text]-invariants of length [Formula: see text] are given by a combination of the Conway polynomials and linking numbers without any vanishing assumption.

HOMFLY polynomials in representation [3, 1] for 3-strand braids

This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3,1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family (n,-1|1,-1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.

On 2-adjacency between links

The author discusses 2-adjacency of two-component links and study the relations between the signs of the crossings to realize 2-adjacency and the coefficients of the Conway polynomial of two related links. By discussing the coefficient of the lowest m power in the Homfly polynomial, the author obtains some results and conditions on whether the trivial link is 2-adjacent to a nontrivial link, whether there are two links 2-adjacent to each other, etc. Finally, this paper shows that the Whitehead link is not 2-adjacent to the trivial link, and gives some examples to explain that for any given two-component link, there are infinitely many links 2-adjacent to it. In particular, there are infinitely many links 2-adjacent to it with the same Conway polynomial.

Legendrian knots and constructible sheaves

We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov-Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in 'a' of the HOMFLY polynomial of the braid closure.

On the link invariants from the Yokonuma–Hecke algebras

In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

Knots, BPS States, and Algebraic Curves

We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture stronger than the known M-theory integrality predictions. Furthermore we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.

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.

On ambiguity in knot polynomials for virtual knots

We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual q and A=qN. These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conveniently observed in the hypercube formalism: then they substitute q-dimensions of certain fat graphs, which are not constrained by recursion and can be chosen arbitrarily. The number of these new topological invariants seems to grow fast with the number of non-virtual crossings: 0, 1, 1, 5, 15, 91, 784, 9160, ... This number can be decreased by imposing the factorization requirement for composites, in addition to topological invariance – still freedom remains. None of these new parameters, however, appears in HOMFLY for Kishino unknot, which thus remains unseparated from the ordinary unknots even by this enriched set of knot invariants.

Knots cascade detected by a monotonically decreasing sequence of values.

Due to reconnection or recombination of neighboring strands superfluid vortex knots and DNA plasmid torus knots and links are found to undergo an almost identical cascade process, that tend to reduce topological complexity by stepwise unlinking. Here, by using the HOMFLYPT polynomial recently introduced for fluid knots, we prove that under the assumption that topological complexity decreases by stepwise unlinking this cascade process follows a path detected by a unique, monotonically decreasing sequence of numerical values. This result holds true for any sequence of standardly embedded torus knots T(2, 2n + 1) and torus links T(2, 2n). By this result we demonstrate that the computation of this adapted HOMFLYPT polynomial provides a powerful tool to measure topological complexity of various physical systems.

Are there p-adic knot invariants?

We suggest to use the Hall-Littlewood version of Rosso-Jones formula to define the germs of $p$-adic HOMFLY-PT polynomials for torus knots $[m,n]$, which possess at least the $[m,n] \longleftrightarrow [n,m]$ topological invariance. This calls for generalizations to other knot families and is a challenge for several branches of modern theory.

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.

Application of Braiding Sequences. II. Polynomial Invariants of Positive Knots

AbstractWe apply the concept of braiding sequences to link polynomials to show polynomial growth bounds on the derivatives of the Jones polynomial evaluated on S1 and of the Brandt–Lickorish–Millett–Ho polynomial evaluated on [–2, 2] on alternating and positive knots of given genus. For positive links, boundedness criteria for the coefficients of the Jones, HOMFLY and Kauffman polynomials are derived. (This is a continuation of the paper ‘Applications of braiding sequences. I’: Commun. Contemp. Math.12(5) (2010), 681–726.)

Invariance and the knot Floer cube of resolutions

This paper considers the invariance of knot Floer homology in a purely algebraic setting, without reference to Heegaard diagrams, holomorphic disks, or grid diagrams. We show that (a small modication of) Ozsváth and Szabó’s cube of resolutions for knot Floer homology, which is assigned to a braid presentation with a basepoint, is invariant under braid-like Reidemeister moves II and III and under conjugation. All moves are assumed to happen away from the basepoint. We also describe the behavior of the cube of resolutions under stabilization. The techniques echo those employed to prove the invariance of HOMFLY-PT homology by Khovanov and Rozansky, and are further evidence of a close relationship between the theories. The key idea is to prove categoried versions of certain equalities satisfied by the Murakami–Ohtsuki–Yamada state model for the HOMFLY-PT polynomial.

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.

On the HOMFLY polynomial of 4-plat presentations of knots

Abstract. In this paper, I give a method to calculate the HOMFLY polynomials of twobridge knots by using a representation of the braid group B 4 into a group of 3×3 matrices.Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same Jonespolynomial, but different HOMFLY polynomials. 1. IntroductionIn 1985, Hoste-Ocneanu-Millett-Freyd-Lickorish-Yetter [3] had discovered the HOMFLYpolynomial which is a 2-variable oriented link polynomial P L (a,m) motivated by the Jonespolynomial. Also, Prztycki and Traczyk [7] independently discovered the HOMFLY polyno-mial. The calculation of the HOMFLY polynomial is based on the HOMFLY skein relationsas follows.(1) P(L) is an isotopy invariant.(2) P(unknot)=1.(3) a·P(L + ) +a −1 ·P(L − )+m·P(L 0 ) = 0.L L _ L+ 0 Figure 1.Actually, Lickorish and Millett [6] gave a formula to calculate the HOMFLY polynomials ofrational knots by using a representation of the continued fraction of a rational knot into agroup of 2× 2 matrices. (See Proposition 14.)In this paper, I use the Kauffman blacket polynomial and the plat presentation of knots tocalculate the HOMFLY polonomials of rational knots by using a representation of the braidgroup B

On universal knot polynomials

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel’s plane, respectively and give their exceptional group’s counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel’s plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

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.

Markov Trace on the Algebra of Braids and Ties

We prove that the so–called algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones recipe we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant for classical knots is an extension of the Homflypt polynomial and the invariant for singular knots is an extension of an invariant of singular knots defined by the second author and S. Lambropoulou.

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

We develop several applications of the fact that the Yokonuma–Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori–Hecke algebras of type A . This includes a description of the semisimple and modular representation theory of the Yokonuma–Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism. In particular, for classical knots, a consequence of the construction is that the obtained set of invariants is topologically equivalent to the HOMFLYPT polynomial. We thus recover results of Chlouveraki et al. (2015, arXiv:1505.06666) about the Juyumaya–Lambropoulou invariants.