Khovanov-Rozansky Homology Research Articles

Categorified Young symmetrizers and stable homology of torus links

We show that the triply graded Khovanov-Rozansky homology of the torus link $T_{n,k}$ stablizes as $k\to \infty$. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes $P_n$ of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that $P_n$ is a stable limit of Rouquier complexes. A certain derived endomorphism ring of $P_n$ computes the aforementioned stable homology of torus links.

Categorified skew Howe duality and comparison of knot homologies

In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin–Turaev invariants for sln. Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category 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 glℓ for some ℓ. In this paper, we show that the construction of knot homology based on categorifying tensor products (from earlier work of the second author) fits into this framework, and thus agrees with other such homologies, such as Khovanov–Rozansky homology. We accomplish this by categorifying the action of glℓ×gln on ⋀p(Cℓ⊗Cn) using diagrammatic bimodules. In this action, the functors corresponding to glℓ and gln are quite different in nature, but they will switch roles under Koszul duality.

Sutured annular Khovanov-Rozansky homology

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Howe duality. This framework then extends to give our annular sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sl(n), which in the n=2 case recovers a result of Grigsby-Licata-Wehrli.

Torus link homology and the nabla operator

In recent work, Elias and Hogancamp develop a recurrence for the Poincaré series of the triply graded Khovanov–Rozansky homology of certain links, one of which is the (n,n) torus link. In this case, Elias and Hogancamp give a combinatorial formula for this homology that is reminiscent of the combinatorics of the modified Macdonald polynomial eigenoperator ∇. We give a combinatorial formula for the homologies of all complexes considered by Elias and Hogancamp. Our first formula is not easily computable, so we show how to transform it into a computable version. Finally, we conjecture a direct relationship between the (n,n) torus link case of our formula and the symmetric function ∇p1n.

Categorified Young symmetrizers and stable homology of torus links II

We construct complexes $$P_{1^{n}}$$ of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture (Flag Hilbert schemes, colored projectors and Khovanov–Rozansky homology. arXiv:1608.07308 , 2016) of Gorsky–Neguț–Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for $$P_{1^{n}}$$ and its twisted variants. We also show that this homology is also a certain limit of Khovanov–Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest.

Khovanov–Rozansky homology and directed cycles

We determine the cycle packing number of a directed graph using elementary projective algebraic geometry. Our idea is rooted in the Khovanov–Rozansky theory. In fact, using the Khovanov–Rozansky homology of a graph, we also obtain algebraic methods of detecting directed and undirected cycles containing a particular vertex or edge.

The [formula omitted] foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality

We give an elementary construction of colored sln link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between sln 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. The latter answers a question posed in [42].

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.

A family of transverse link homologies

We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N \geq 1$, $\mathcal{H}_N$ is a $\mathbb{Z}_2\oplus\mathbb{Z}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module that is invariant under transverse Markov moves, but not under negative stabilization/de-stabilization. Thus, for $N\geq 1$, this homology is an invariant for transverse links in the standard contact $S^3$, but not for smooth links. We also discuss the decategorification of $\mathcal{H}_N$ and the relation between $\mathcal{H}_N$ and the $\mathfrak{sl}(N)$ Khovanov-Rozansky homology defined in arXiv:math/0401268.

Some differentials on Khovanov–Rozansky homology

We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N > 0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR–homology of knots with 9 crossings or fewer.

Equivariant Khovanov–Rozansky homology and Lee–Gornik spectral sequence

Lobb observed in [8] that each equivariant \mathfrak {sl} (N) Khovanov–Rozansky homology over \mathbb C [a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee–Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alternative definition of the Lee–Gornik spectral sequence using exact couples. We also demonstrate that an equivariant \mathfrak {sl} (N) Khovanov–Rozansky homology over \mathbb C [a] can be recovered from the corresponding Lee–Gornik spectral sequence via this formula. Therefore, these two algebraic invariants are equivalent and contain the same information about the link. As a byproduct of the exact couple construction, we generalize Lee's endomorphism on the rational Khovanov homology to a natural \bigwedge^\ast \mathbb C^{N-1} -action on the \mathfrak sl (N) Khovanov–Rozansky homology. A numerical link invariant called torsion width comes up naturally in our work. It determines when the corresponding Lee–Gornik spectral sequence collapses and is bounded from above by the homological thickness of the \mathfrak {sl} (N) Khovanov–Rozansky homology. We use the torsion width to explain why the Lee spectral sequences of certain H-thick links collapse so fast.

On the Braid Index of Kanenobu Knots

We study the braid indices of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. The MFW inequality is known for giving a lower bound of the braid index of a link by applying the HOMFLYPT polynomial. Therefore, it is not easy to determine the braid indices of the Kanenobu knots. In our previous paper, we gave upper bounds and sharper lower bounds of the braid indices of the Kanenobu knots by applying the 2-cable version of the zeroth coefficient HOMFLYPT polynomial. In this paper, we give sharper upper bounds of the braid indices of the Kanenobu knots.

Torus knots and the rational DAHA

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n−1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov–Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q,t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.

2–strand twisting and knots with identical quantum knot homologies

Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giving a new way to generate arbitrary numbers of knots with isomorphic homologies and finding an infinite number of mutant knot pairs with isomorphic reduced homologies.

Computing Khovanov–Rozansky homology and defect fusion

We compute the categorified sl(N) link invariants as defined by Khovanov and Rozansky, for various links and values of N. This is made tractable by an algorithm for reducing tensor products of matrix factorisations to finite rank, which we implement in the computer algebra package Singular.

THE MODULI PROBLEM OF LOBB AND ZENTNER AND THE COLORED 𝔰𝔩(N) GRAPH INVARIANT

Motivated by a possible connection between the SU (N) instanton knot Floer homology of Kronheimer and Mrowka and 𝔰𝔩(N) Khovanov–Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colorings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yamada in their state-sum interpretation of the quantum 𝔰𝔩(N) knot polynomial. For graphs with two colors, they showed this moduli space can be thought of as a representation variety, and that its Euler characteristic is equal to the 𝔰𝔩(N) polynomial of the graph evaluated at 1. We extend their results to graphs with arbitrary colorings by irreducible anti-symmetric representations of 𝔰𝔩(N).

THE ZEROTH COEFFICIENT HOMFLYPT POLYNOMIAL OF A 2-CABLE KNOT

The zeroth coefficient polynomial is a one variable polynomial contained in the HOMFLYPT polynomial. In this paper, we give a basic computation of the zeroth coefficient polynomial of a 2-cable knot. In particular, we compute the zeroth coefficient polynomials of the 2-cable knots of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov–Rozansky homology. As a result, we distinguish the Kanenobu knots completely. Moreover, we estimate the braid indices of the Kanenobu knots.

A note on Gornik’s perturbation of Khovanov–Rozansky homology

We show that the information contained in the associated graded vector space to Gornik's version of Khovanov-Rozansky knot homology is equivalent to a single even integer s_n(K). Furthermore we show that s_n is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen's invariant coming from a perturbation of Khovanov homology.

Quantum (sln, ∧V n ) link invariant and matrix factorizations

AbstractIn this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where∧Vnis the set of fundamental representations ofUq(sln). In the case of an oriented link diagram composed of[k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general[i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

Quantum (sln, ∧V n) link invariant and matrix factorizations

AbstractIn this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where ∧Vn is the set of fundamental representations of Uq(sln). In the case of an oriented link diagram composed of [k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general [i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.