This article gives matrix factorizations for the trivalent diagrams and double line appearing in $\mathfrak{sl}_n$ quantum link invariant.
These matrix factorizations reconstruct Khovanov-Rozansky homology. And we show that the Euler characteristic of the matrix factorization for a double loop equals the quantum dimension of the representation $\land^2 V$ of $U_q (\mathfrak{sl}_n)$ in Section \ref{sec3.3}.
The $\mathfrak{sl}_3$ colored Jones polynomial $J_λ^{\mathfrak{sl}_3}(L)$ is obtained by coloring the link components with two-row Young diagram $λ$. Although it is difficult to compute $J_λ^{\mathfrak{sl}_3}(L)$ in general, we can calculate it by using Kuperberg's $A_2$ skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in $A_2$ web space and compute $J_{(n,0)}^{\mathfrak{sl}_3}(T(2,2m))$ for an oriented $(2,2m)$-torus link. These explicit formulas derives the $\mathfrak{sl}_3$ tail of $T(2,2m)$. They also give explicit descriptions o…
The colored $\mathfrak{sl}_{3}$ Jones polynomial $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ are given by a link and an $(n_{1}, n_{2})$-irreducible representation of $\mathfrak{sl}_{3}$. In general, it is hard to calculate $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ for an oriented link $L$. However, we calculate the one-row $\mathfrak{sl}_{3}$ colored Jones polynomials $J_{(n, 0)}^{\mathfrak{sl}_{3}}(P(α,β,γ);q)$ for three-parameter families of oriented pretzel links $P(α,β,γ)$ by using Kuperberg's linear skein theory by setting $n_{2}=0$. Furthermore, we show the existence of the tails …
We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.
We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivariant cohomology rings of partial flag manifolds.
For an $m$-periodic link $L$, we show that the Khovanov-Rozansky $\mathfrak{sl}_N$-homology carries an action of the group $\mathbb{Z}_m$. As an example of applications, we prove an analog of the periodicity criterion of Borodzik--Politarczyk using $\mathfrak{sl}_N$-homology instead of Khovanov homology.
Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin's "$\mathfrak{sl}(n)$-like" Heegaard Floer knot invariants $HFK_n$ recover both Alexander polynomial evaluations and $\mathfrak{sl}(n)$ polynomial evaluations at certain roots of unity for links in $S^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $\mathfrak{sl}(n)$ homology and $HFK_n$.
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…
To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.
Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ colored Jones polynomials of $2$-bridge knots and links explicitly using twist formulas.
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.
We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of $\mathfrak{sl}_N$. In addition, the construction is actually made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.
In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $β_3$-invariants) which are transverse braid invariants. This family includes Wu's $ψ_{3}$-invariant. Furthermore, we analyse the vanishing of the homology classes of the $β_3$-invariants and relate it to the vanishing of Plamenevskaya's and Wu's invariants. Finally, we use the $β_3$-invariants to produce some Bennequin-type inequalities.
The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathfrak{sl}_3$ colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the $\mathfrak{sl}_3$ colored Jones polynomials colored by $(n,0)$ for the $(2,2m)$-torus link. These two expressions of the tail provide an identity of $q$-seri…
We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projective modules of $\overline{U}_q^H(\mathfrak{sl}_2)$. These relate to the colored Alexander tangle invariants studied in [ADO, M1]. It follows that the regularized asymptotic dimensions of characters of $\mathcal M(r)$ coincide with the corresponding modified traces of open Hopf lin…
We give a purely combinatorial construction of colored $\mathfrak{sl}_n$ link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between $\mathfrak{sl}_n$ webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov-Rozansky homology. Novel features of the theory include the introduction of `enhanced' foam facets which fix sign issues associated with the original matrix factorization formulation and the use of skew Howe duality to show that (enhanced) closed foams can be evaluated in a completely combinatorial manner…
In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd order. These constructions use the notion of modified trace and relative $\mathit{G}$-modular category.
We present state sums for quantum link invariants arising from the representation theory of $U_q(\mathfrak{gl}_{N|M})$. We investigate the case of the $N$-th exterior power of the standard representation of $U_q(\mathfrak{gl}_{N|1})$ and explicit the relation with Kashaev invariants.
We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield $\mathfrak{gl}_{-n}$ link homology, i.e. a link homology theory associated to the Lie superalgebra $\mathfrak{gl}_{0|n}$, both for links in $S^3$ and in the thickened annulus. In the $n=2$ case, this produces a categorification of the Jones polynomial that we show is distinct from Khovanov homology, and gives a finite-dimensional categorification of the colored Jones polyn…
We introduce a generalization of the Lisca-Ozsváth-Stipsicz-Szabó Legendrian invariant $\mathfrak L$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link $L$ in a contact 3-manifold $(M,ξ)$ with a diagram $D$, given by an open book decomposition of $(M,ξ)$ adapted to $L$, and we construct a chain complex $cCFL^-(D)$ with a special cycle in it denoted by $\mathfrak L(D)$. Then, given two diagrams $D_1$ and $D_2$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complex…
We develop a diagrammatic calculus for representations of unrolled quantum $\mathfrak{sl}_2$ at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given here are a skein relation for $n$-cabled double crossings and a simple proof that the quantum invariant associated with these representations determines the multivariable Alexander polynomial.
We show that the reduced $\mathrm{SL}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $\mathcal{U}_q(\mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $\operatorname{SL}_2(\mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their in…
We define parameter dependent $\mathfrak{gl}_2$-foams and their associated web and arc algebras, and verify that they specialize to several known $\mathfrak{sl}_2$ or $\mathfrak{gl}_2$ constructions related to higher link and tangle invariants. Moreover, we show that all these specializations are equivalent, and we deduce several applications, e.g. we discuss the consequences for the associated link and tangle invariants, and their functoriality.
In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the $\mathfrak{gl}_N$ Reshetikhin--Turaev link invariants and the universal $\mathfrak{gl}_N$ knot invariant; 2) an explicit construction of the unified $\mathfrak{gl}_N$ invariants for integral homology 3-spheres using universal Kirby colors. These results …
The Reshetikhin-Turaev sl(N) polynomial of links colored by wedge powers of the defining representation has been categorified via several different approaches. Here, we give a concise introduction to the categorification using matrix factorizations, which is a direct generalization of the Khovanov-Rozansky homology. Full details of the construction are given in [arXiv:0907.0695]. We also briefly review deformations and applications of this categorification given in [arXiv:1002.2662, arXiv:1011.2254, arXiv:1102.0586].
We define and study a family of link invariants $\mathit{HFK}_{n}(L)$. Although these homology theories are defined using holomorphic disc counts, they share many properties with $sl_{n}$ homology. Using these theories, we give a framework that generalizes the conjectured spectral sequence from Khovanov homology to $δ$-graded knot Floer homology. In particular, we conjecture that for all links $L$ in $S^3$ and all $n\ge 1$, there is a spectral sequence from the $sl_{n}$ homology of $L$ to $\mathit{HFK}_{n}(L)$.
The contents of this 98-page paper have been subsumed into the 191-page paper "A colored sl(N)-homology for links in S^3" (arXiv:0907.0695v1 [math.GT]), in which we further develop the theory and use it to construct a colored link homology.
We modify our previous construction of link homology in order to include a natural duality functor $\mathfrak{F}$. To a link $L$ we associate a triply-graded module $HXY(L)$ over the graded polynomial ring $R(L)=\mathbb{C}[x_1,y_1,\dots,x_\ell,y_\ell]$. The module has an involution $\mathfrak{F}$ that intertwines the Fourier transform on $R(L)$, $\mathfrak{F}(x_i)=y_i$, $\mathfrak{F}(y_i)=x_i$. In the case when $\ell=1$ the module is free over $R(L)$ and specialization to $x=y=0$ matches with the triply-graded knot homology previously constructed by the authors. Thus we show that the correspon…
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 $\mathbb{R}^3$, as a sum over BPS webs attached to the link.
We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated and regenerated surfaces in the latter. In particular we focus on degenerations which involve intersection points of multiplicity two and three. We demonstrate when the same links arise even when the local arrangements are different.
J.P. Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the $\barμ$-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ${\Bbb Z}[t,t^{-1}]$. In addition, we give a relation between the Taylor expansion of a l…
In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for $\mathfrak{sl}_n$. Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category $\mathcal{O}$, affine Grassmannians, and diagrammatic categorifications of tensor products. While the definitions of these theories are quite different, there is a key commonality between them which makes it possible to prove that they are all isomorphic: they arise from a skew Howe dual action of $\mathfrak{gl}_\ell$ for some $…
Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Natário and Tod says that two events $x,y\inß$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial in the contact manifold of all light geodesics in $X$. The Low conjecture says that for $m=2$ the events $x,y$ are causally related if and only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological l…
A string link S can be closed in a canonical way to produce an ordinary closed link L. We also consider a twisted closing which produces a knot K. We give a formula for the Conway polynomial of L as a product of the Conway polynomial of K times a power series whose coefficients are given as explicit functions of the Milnor invariants of S. One consequence is a formula for the first non-vanishing coefficient of the Conway polynomial of L in terms of the Milnor invariants of L. There is an analogous factorization of the multivariable Alexander polynomial.
For every knot $K$ and lie algebra $\mathfrak{g}$, there is a Gukov-Manolescu series denoted $F^{\mathfrak{g}}_K$ which serves as an analytic continuation of the quantum knot invariants associated to finite dimensional irreducible representations of $\mathfrak{g}$. There has been a great deal of work done on computing this invariant for $\mathfrak{g} = \mathfrak{sl}_2$ but comparatively less work has studied other lie algebras. In this paper we extend the large colour $R$ matrix from $\mathfrak{sl}_2$ to symmetrically coloured $\mathfrak{sl}_N$. This gives a definition for $F^{\mathfrak{sl}_N,…
Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra $\mathfrak{sl}_2$, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of $\mathfrak{sl}_2$ are lifted to maps of spectra. In particular, it follows that the $\mathfrak{sl}_2$ action on homology commutes with the action of the Steenrod algebra. The main new technical ingredients developed in this paper, which may be …
We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $\mathfrak{T}$. We also give a diagrammatic category for the (graded) projective endofunctors on $\mathfrak{T}$, indicate how our results could generalize and collect some "well-known" facts to give a reasonably self-contained exposition.
We study webs in quantum type $C$, focusing on the rank three case. We define a linear pivotal category $\mathbf{Web}(\mathfrak{sp}_6)$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\mathbf{FundRep}(U_q(\mathfrak{sp}_6))$ of quantum $\mathfrak{sp}_6$ representations generated by the fundamental representations, for generic values of the parameter $q$. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\mathbf{Web}(\mathfrak{sp}_6) \rightarrow \mathbf{FundRep}(U_q(\m…
These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triply graded link homology. The homology is space of global sections of a particular sheaf on the Hilbert scheme of points on the plane. Our construction relies on existence on the natural push-forward functor for the equivariant matrix factorizations, we explain the subtleties on the construction in these notes. We also outline a proof of the Markov moves for our homology as well as some explicit localizati…
