HOMFLY Research Articles

Planar decomposition of the HOMFLY polynomial for bipartite knots and links

AbstractThe theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from $$\mathfrak {sl}(2)$$ sl ( 2 ) to $$\mathfrak {sl}(N)$$ sl ( N ) at arbitrary N – but for a special class of bipartite diagrams made entirely from the antiparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters – and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov–Rozansky generalizations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

Factors of HOMFLY polynomials

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs.

Harer–Zagier transform of the HOMFLY–PT polynomial for families of twisted hyperbolic knots

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived expressions for the Harer–Zagier transform—a discrete Laplace transform—of the HOMFLY–PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of pretzel knots for which, like for torus knots, the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zero loci show a remarkable structure, which mostly lies on the unit circle and deviates from it only in pairs.

U(N) torus link invariants in the large N limit from the matrix model approach

In this paper we study U(N) colored HOMFLY-PT polynomials of torus links in the double scaling limit (polynomial variable q→1, N→∞ keeping qN fixed). We show that, in this limit, the colored HOMFLY-PT polynomial of any (Lα,Lβ) torus link can be expressed in terms of the colored HOMFLY-PT polynomial of (L,L) torus link. Using the connection between matrix models and the Chern-Simons field theoretic invariants, we show that the colored torus link invariants are uniquely expressed in terms of connected correlation functions of operators in U(N) matrix model. We determine the leading and subleading contribution to some of the correlators at large N from the matrix model approach and find that they match exactly with those obtained from the corresponding colored HOMFLY-PT polynomials. Published by the American Physical Society 2024

Olympic gels formed through catenation of dsDNA rings regulated by topoisomerase II: A coarse-grained model.

Topological regulation of DNA by topoisomerases in cells is very crucial for life. We propose a coarse-grained model to study the catenation process of double-stranded DNA (dsDNA) rings regulated by topoisomerase II (TOP2) and provide a computational method to characterize the topological structures of the Olympic gels obtained. The function of TOP2 in the catenation of dsDNA rings is implicitly fulfilled by operating the length of a stretchable catch bond in the dsDNA ring. After the catenation reaction of initially noncatenated dsDNA rings in the solution, the Olympic gel is obtained and the interlocked topology of the dsDNA rings can be characterized by a computational method derived from the HOMFLY polynomial, based on which the catenation degree and the complexity of catenation are quantified. Detailed dependence of the catenation degree and the complexity of the catenated topology on key parameters, including the size of the transient broken gap and the duration time of the break on the dsDNA ring during operation by TOP2, the initial molar ratio of TOP2 to the dsDNA rings, and the reaction temperature, has been investigated.

The HOMFLY Polynomials of Class of Links

The HOMFLY Polynomials of Class of Links

Knot homologies and generalized quiver partition functions

We introduce generalized quiver partition functions of a knot K and conjecture a relation to generating functions of symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials. We interpret quiver nodes as certain basic holomorphic disks in the resolved conifold, with boundary on the knot conormal LK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_K$$\\end{document}, a positive multiple of a unique closed geodesic, and with their (infinitesimal) boundary linking density measured by the adjacency matrix of the generalized quiver. The basic holomorphic disks that are quiver nodes appear in a certain U(1)-symmetric configuration. We propose an extension of the quiver partition function to arbitrary, not U(1)-symmetric, configurations as a function with values in chain complexes. The chain complex differential is trivial at the U(1)-symmetric configuration, under deformations the complex changes, but its homology remains invariant. We also study recursion relations for the partition functions connected to knot homologies. We show that, after a suitable change of variables, any (generalized) quiver partition function satisfies the recursion relation of a single toric brane in C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}^3$$\\end{document}.

Gauss diagram formulae for Vassiliev invariants from Kauffman polynomial

A state model for Kauffman polynomial of Dubrovnik-version is described using Gauss diagrams. Based on the state model, the Gauss diagram formulae for Vassiliev invariants are given from the coefficients of Kauffman polynomial following the method of Chmutov and Polyak. Some arrow diagram identities are given to simplify the Gauss diagram formulae of order 3, which give Polyak–Viro and Chmutov–Polyak formulae for the Vassiliev invariant of order 3. The models of Kauffman polynomial and HOMFLY-PT polynomial give different Gauss diagram expressions when specializing in Jones polynomial.

HOMFLY-PT polynomials of open links

We numerically estimate the superposition of the HOMFLY-PT polynomial of an open two-component link, define its spread, and describe how this quantity may be employed to quantify the degree of entanglement of confined two component open links.

The quantum UV-IR map for line defects in mathfrak{gl} (3)-type class S theories

We consider the quantum UV-IR map for line defects in class S theories of mathfrak{gl} (3)-type. This map computes the protected spin character which counts framed BPS states with spin for the bulk-defect system. We give a geometric method of computing this map motivated by the physics of five-dimensional mathcal{N} = 2 supersymmetric Yang-Mills theory, and compute it explicitly in various examples. As a spin-off we propose a new way of computing a certain specialization of the HOMFLY polynomial for links in ℝ3, as a sum over BPS webs attached to the link.

Evolution properties of the knot’s defect

The defect of differential (cyclotomic) expansion for colored HOMFLY-PT polynomials is conjectured to be invariant under any antiparallel evolution and change linearly with the evolution in any parallel direction. In other words, each {{mathcal {R}}}-matrix can be substituted by an entire 2-strand braid in two different ways: the defect remains intact when the braid is antiparallel and changes by half of the added length when the braid is parallel.

A new polynomial criterion for periodic knots

The purpose of this paper is to present a new periodicity criterion. For that purpose, we study the HOMFLY-PT polynomial and the Kauffman polynomial of cables of periodic links. Furthermore, we exhibit a couple of examples for which our criterion is stronger than many previously known criteria, among which is the Khovanov homology criterion of Borodzik and the second author.

$${\widehat{Z}}$$ at Large N: From Curve Counts to Quantum Modularity

Reducing a 6d fivebrane theory on a 3-manifold $Y$ gives a $q$-series 3-manifold invariant $\widehat{Z}(Y)$. We analyse the large-$N$ behaviour of $F_K=\widehat{Z}(M_K)$, where $M_K$ is the complement of a knot $K$ in the 3-sphere, and explore the relationship between an $a$-deformed ($a=q^N$) version of $F_{K}$ and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of $F_K$ in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for $a$-deformed $F_K$ for $(2,2p+1)$-torus knots. They suggest a further $t$-deformation based on superpolynomials, which can be used to obtain a $t$-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how $F_K$ transforms under natural geometric operations on $K$ indicates relations to quantum modularity in a new setting.

Colored HOMFLY-PT polynomials of quasi-alternating 3-braid knots

Obtaining a closed form expression for the colored HOMFLY-PT polynomials of knots from 3-strand braids carrying arbitrary SU(N) representation is a challenging problem. In this paper, we confine our interest to twisted generalized hybrid weaving knots which we denote hereafter by Qˆ3(m1,−m2,n,ℓ). This family of knots not only generalizes the well-known class of weaving knots but also contains an infinite family of quasi-alternating knots. Interestingly, we obtain a closed form expression for the HOMFLY-PT polynomial of Qˆ3(m1,−m2,n,ℓ) using a modified version of the Reshitikhin-Turaev method. In addition, we compute the exact coefficients of the Jones polynomials and the Alexander polynomials of quasi-alternating knots Qˆ3(1,−1,n,±1). For these homologically-thin knots, such coefficients are known to be the ranks of their Khovanov and link Floer homologies, respectively. We also show that the asymptotic behavior of the coefficients of the Alexander polynomial is trapezoidal. On the other hand, we compute the [r]-colored HOMFLY-PT polynomials of quasi alternating knots for small values of r. Remarkably, the study of the determinants of certain twisted weaving knots leads to establish a connection with enumerative geometry related to mth Lucas numbers, denoted hereafter as Lm,2n. At the end, we verify that the reformulated invariants satisfy Ooguri-Vafa conjecture and we express certain BPS integers in terms of hyper-geometric functions F12[a,b,c;z].

A note on HOMFLY polynomial of positive braid links

For a positive braid link, a link represented as a closed positive braid, we determine the first few coefficients of its HOMFLY polynomial in terms of geometric invariants such as, the maximum Euler characteristics, the number of split factors, and the number of prime factors. Our results give improvements of known results for the Conway and the Jones polynomial of positive braid links. In Appendix, we present a simpler proof of theorem of Cromwell, a positive braid diagram represents a composite link if and only if the diagram is composite.

On the HOMFLY polynomials of even trigonal bipyramid links

On the HOMFLY polynomials of even trigonal bipyramid links

Topological recursion for the extended Ooguri–Vafa partition function of colored HOMFLY-PT polynomials of torus knots

We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).

Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU(N) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots T[2,2k+1]. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank N and the representation R.

Khovanov polynomials for satellites and asymptotic adjoint polynomials

In this paper, we compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companion invariants. For a given companion, the satellite polynomial “smoothly” depends on the pattern but for the “jump” at one critical point defined by the [Formula: see text]-invariant of the companion knot. A similar phenomenon is known for the knot Floer homology and [Formula: see text]-invariant for the same kind of satellites.

Mathematical Logic of the Jones and Homfly Polynomials of Knotted Trivalent Networks

Two rational functions are defined logically for special type of knotted trivalent networks as state models of planar trivalent networks. The restriction of these two rational functions reduce to the Jones and Hom y polynomials for non oriented links. Also, these two models are used to define two invariants for this special type of knotted trivalent networks embedded in R3. Finally, we study some congruences of these two polynomials for periodic knotted trivalent networks this generalize the work of periodicity of the Jones and Hom y polynomials on knots to these two rational functions of knotted trivalent networks.