Hoel Queffelec, Antonio Sartori
We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY-PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the gl(m|n) quantum invariants) as link invariants with values in the quantized oriented Brauer category.
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In this paper we define the 1,2-coloured HOMFLY-PT link homology and prove that it is a link invariant. We conjecture that this homology categorifies the coloured HOMFLY-PT polynomial for links whose components are labelled 1 or 2.
This note is a write-up of a talk given by the author at the Meeting of the Sociedade Portuguesa de Matematica in July 2012. We describe Jaeger's HOMFLY-PT expansion of the Kauffman polynomial and how to generalize it to other quantum invariants using the so-called "branching rules" for Lie algebra representations. We present a program which aims to construct Jaeger expansions for link homology theories.
We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.
Using the diagrammatic calculus for Soergel bimodules, developed by B. Elias and M. Khovanov, as well as Rasmussen's spectral sequence, we construct an integral version of HOMFLY-PT and sl(n)-link homology.
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reidemeister moves. In this text we prove that the Reidemeister IIb move fails in HOMFLY-PT homology by using virtual crossing filtrations of the author and Rozansky. The decategorification of HOMFLY-PT homology for general link diagrams gives a deformed version of the HOMFLY-PT poly…
The HOMFLY polynomial for torus links from Chern-Simons gauge theory
在Chern-Simons规范理论的框架下，利用纽结算子计算了任意环面纽结和链环的规范群$SU(N)$的基本表示所对应的多项式不变量.最后给出了任意环面连杆的HOMFLY多项式公式。
The HOMFLY polynomial for links in rational homology 3-spheres
我们构造了一个多项式不变量，链接在塞弗特纤维或环面有理同调3-球面，推广了2-变量琼斯多项式（HOMFLY）。因此，我们证明了同伦3-球面的HOMFLY skein模的对偶与真3-球面的HOMFLY skein模同构。
We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with one-column (or one-row) Young diagrams.
We explain how existing results (such as categorical sl(n) actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the HOMFLY polynomial of links coloured by arbitrary partitions. The construction uses a categorified HOMFLY clasp defined via cabling and infinite twists. We briefly discuss differentials and speculate on related structures.
We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.
HOMFLYPT homology for links in handlebodies via type A Soergel bimodules
本文利用经典辫群子群与奇异Soergel双模的范畴作用，将3-球中着色HOMFLYPT同调推广为亏格g带柄体中链环的三重分次不变量。
In this paper, we present several new structures for the colored HOMFLY-PT invariants of framed links. First, we prove the strong integrality property for the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein theory developed by H. Morton and his collaborators. By this strong integrality property, we immediately obtain several symmetric properties for the full colored HOMFLY-PT invariants of links. Then, we apply our results to refine the mathematical structures appearing in the Labastida-Mariño-Ooguri-Vafa (LMOV) integrality conjecture for framed links. As another ap…
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulated) colored HOMFLY-PT invariants actually lie in the ring $\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Secondly, we establish some symmetric formulas for colored HOMFLY-PT invariants of links, which include the rank-level duality as an easy consequence. Finally, motivated by the Labastid…
On the Hecke algebras and the colored HOMFLY polynomial
本文利用缆化-投影规则推导出有色HOMFLY多项式用Hecke代数特征标和Schur多项式表示的公式，并由此得到环面链环的简洁公式，从而可在环面链环上检验揭示Chern-Simons规范理论与弦理论深层联系的Labastida-Mariño-Vafa猜想。
A HOMFLYPT-type invariant for pseudo links via a resolution in Hecke algebras
本文通过Hecke代数中的解析构造了伪链接的HOMFLYPT型不变量，解决了伪 Reidemeister 1 移动带来的障碍，基于解析同态和Ocneanu迹得到自然的伪 skein 关系。
With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g+1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU_q(N) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many muta…
We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Stošić about the growth of colored HOMFLY-PT homologies.
In this note we describe the recursion relations between two parameter HOMLFY and Kauffman polynomials of framed links These relation correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings $g_{n}\supset g_{k}\times sl_{n-k}$ where $g_{n}$ is either $so_{2n+1}$, $so_{2n}$ or $sp_{2n}$ is new.
We prove that the generating functions for the one row/column colored HOMFLY-PT invariants of arborescent links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. Our approach extends the previously established tangles-quivers correspondence for rational tangles to algebraic tangles by developing gluing formulas for HOMFLY-PT skein generating functions under Conway's tangle addition. As a consequence, we prove the conjectural links-quivers correspondence of Kucharski-Reineke-Stošić-Sulk…
We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a,q) super-polynomial of all knots in 3-space. Our result has implications on the quantization of the SL(2,C) character variety of knots using ideal triangulations or the topological recursion, and motivates questions on the web approach to representation theory.
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformation of toric conformal block.
This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah.
We give an exposition of how the Kauffman bracket arises for certain systems of anyons, and do so outside the usual arena of Temperley-Lieb-Jones categories. This is further elucidated through the discussion of the Iwahori-Hecke algebra and its relation to modular tensor categories. We then proceed to classify the framed link-invariants associated to a system of self-dual anyons $q$ with $\sum_{x}N_{qq}^{x}\leq2$. In particular, we construct a trace on the HOMFLY skein algebra which can be expanded via gauge-invariant quantities, thereby generalising the case of the Kauffman bracket. Various e…
We consider vector spaces H(n,l) and F(n,l) spanned by the degree-n coefficients in power series forms of the Homfly and Kauffman polynomials of links with l components. Generalizing previously known formulas, we determine the dimensions of the spaces H(n,l), F(n,l) and H(n,l)+F(n,l) for all values of n and l. Furthermore, we show that for knots the algebra generated by H(n,1)+F(n,1) (n > 0) is a polynomial algebra with dim(H(n,1)+F(n,1))-1=n+[n/2]-4 generators in degree n>3 and one generator in degrees 2 and 3.
The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur--Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials, and discuss extensions of these quantum objects designed to encompass all of the mentioned invariants, including the HO…
We compute the vacuum expectation values of torus knot operators in Chern-Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus links and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.
Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of arXiv:1304.5778 that the large representation limit (the same as considered in the knot volume conjecture) of this quantity matches the prediction from mirror symmetry consideration. We also provide, using the evolution method, the HOMFLY polynomial in two arbitrary symmetric repres…
In this paper, we study the quantum $\mathfrak{sl}(n)$ representation category using the web space. Specially, we extend $\mathfrak{sl}(n)$ web space for $n\ge 4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of $\mathfrak{sl}(n)$. Moreover, we correct the false conjecture \cite{PS:superiod} given by Chbili, which addresses the relation between some link polynomials of a periodi…
Doubly-graded exponential growth of colored torus knot homology
本研究通过构造彩色扭结的HOMFLY同调，证明了其双格不变量在颜色上的指数增长，并验证了相关猜想。
We suggest to associate with each knot the set of coefficients of its HOMFLY polynomial expansion into the Schur functions. For each braid representation of the knot these coefficients are defined unambiguously as certain combinations of the Racah symbols for the algebra SU_q. Then, the HOMFLY polynomials can be extended to the entire space of time-variables. The so extended HOMFLY polynomials are no longer knot invariants, they depend on the choice of the braid representation, but instead one can naturally discuss their explicit integrable properties. The generating functions of torus knot/li…
We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k) representations. Projective functors restricted to these categories give rise to a functorial action of the Lie algebra. On the derived category, Zuckerman functors categorify sl(k)- homomorphisms. Cones of natural transformations between the identity functor and Zuckerman functors are assigned to crossings and this assignment satisfies the appropriate relations. On the Grothendieck group, the funct…
Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from 3-strand braid, can be constructed using the quantum Racah coefficients (6j-symbols) of $U_q(sl_2)$. In this paper, we generalise it to links whose components carry different symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials $H_{[r_1],[r_2]}$ for th…
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 hold for 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.
Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first …
This paper describes a method for the automatic evaluation of the Links-Gould two-variable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at most 5. Data are presented for the invariant, for all prime knots of up to 10 crossings and various other links. LG distinguishes between these links, and also detects the chirality of those that are chiral. In this sense, it is more sensitive than the well-known two-variable HOMFLY and Kauffman polynom…
This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in various representations a la arXiv:1112.2654, (iii) tabulating and collecting the results at www.knotebook.org. In this paper we discuss only representation R=[2,1] and construct all necessary ingredients that allow one to evaluate knot/links represented by three strand closed paral…
For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.
We define a ribbon category $\mathsf{Sp}(β)$, depending on a parameter $β$, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $β=m-n$ the monoidal category of representations of $U_q(\mathfrak{gl}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(β)$ with a direct limit of quotients of a dual idempotented quantum group $\dot{\mathsf{U}}_q(\mathfrak{gl}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the …
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's $G_2$ quantum invariants. Our method consists in the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.
Following the recent work by Chan, and by Morton and Hadji on the Homflypt polynomials of some generalized Hopf links, we investigate the Kauffman polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S^1\times D^2, we establish a similar skein map on the Kauffman skein module of S^1\times D^2 which has distinct eigenvalues. Furthermore we are able to calculate the Kauffman polynomials of some specific generalized Hopf links.
A Geometric Approach to the Links-Quivers Correspondence I: Rational Tangles
本文通过几何方法证明了有理结的链接-有向图对应关系，利用绕数和配置空间描述了对应的有向图结构。
Panhandle polynomials of torus links and geometric applications
本文通过量子群和罗斯-琼斯公式，研究拓扑链接的环形多项式性质，并探讨其在几何应用中的扩展。
In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov-Rozansky graph homology.
The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant of oriented (1,1) tangles, which is derived from the representation of the braid generator associated with the one-parameter family of four dimensional representations with highest weights (0,0|a) of the quantum superalgebra U_q[gl(2|1)]. We use an abstract tensor state model to evaluate the invariant, as per the construction of the bracket polynomial state model used by Louis Kauffman to derive the Jones polynomial. This model facilitates both computation and theoretical exploration. Our family of representations has…
M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum $(sl_n,\land V_n)$ link invariant, where $\land V_n$ is the set of the fundamental representations of the quantum group of $sl_n$. In the case of a [1,k]-colored link diagram, we prove that its homology is a link invariant. In the case of an [i,j]-colored link diagram, we define a normalized Poincare polynomial of its homology and prove the polynomial is a link invariant.
The usual construction of link invariants from quantum groups applied to the superalgebra D_{2 1,alpha} is shown to be trivial. One can modify this construction to get a two variable invariant. Unusually, this invariant is additive with respect to connected sum or disjoint union. This invariant contains an infinity of Vassiliev invariants that are not seen by the quantum invariants coming from Lie algebras (so neither by the colored HOMFLY-PT nor by the colored Kauffman polynomials).
We extend the construction of the DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type A from the iterated torus knots (our previous paper) to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is directly connected to the so-called splice diagrams. The specialization t=q of our superpolynomials conjecturally results in the HOMFLY-PT polynomials. The relation of our construction to the stable Khovanov-Rozansky polynomials and the so-called ORS-polynomials of the correspond…
The quantum G_2 link invariant
我们推导出一个归纳的，组合定义的多项式值定期同痕不变量的链接和纠缠图。证明了该不变量等于例外单李代数G_2对应的Reshetikhin-Turaev不变量。因此，它与G_2的关系就像HOMFLY多项式与A_n的关系以及Kauffman多项式与B_n、C_n和D_n的关系一样。我们给出了其他秩2李代数的平行构造，并给出了一些基于新归纳定义的组合构造。
We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.
Computing the Khovanov homology of 2 strand braid links via generators and relations
本研究拓展Murakami等人的相关构造，通过组合定义的有向三价平面图形式复形等价类范畴表示Khovanov-Rozansky同调，规避矩阵分解计算难题，实现任意参数下2股辫链环的n级Khovanov-Rozansky不变量计算，证实并拓展多项先前结果与猜想。
On detection probabilities of link invariants
研究证明标准链不变量检测交替链的概率随n呈指数衰减，并通过大数据方法验证了这一现象在结中的普遍性。
The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conifold with four different representations on the four external legs, on the topological string side, is described by a special projection of the four-component link $L_{8n8}$, which reduces to the Hopf link colored with two composite representations. Thus, this provides the first e…
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditio…
We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for every closed, framed 3-manifold M. When H is a quantized universal enveloping algebra, #(M,H) is closely related to well-known quantum link invariants such as the HOMFLY polynomial, but it is not a topological quantum field theory.
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a $O(N^{\frac{3}{2} \mathrm{cw}} \mathrm{poly}(n))$ time algorithm to compute any Reshetikhin-Turaev invariant---derived from a simple Lie algebra $\mathfrak{g}$---of a link presented by a planar diagram with $n$ crossings and carving-width $\mathrm{cw}$, and whose components are coloured with $\mathfrak{g}$-modules of dimension at most $N$. For example, this incl…
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