Link Homology Research Articles

2-block Springer fibers: convolution algebras and coherent sheaves

For a fixed 2-block Springer fiber, we describe the structure of its irreducible components and their relation to the Białynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in \mathbb{C}^n preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the action of diagonal matrices commuting with our fixed nilpotent. In particular, we describe the structure of each component, its set of torus fixed points, and prove a conjecture of Fung describing the intersection of any pair. Then we define a convolution algebra structure on the direct sum of the cohomologies of pairwise intersections of irreducible components and closures of \mathbb{C}^* -attracting sets (that is Białynicki-Birula cells), and show this is isomorphic to a generalization of the arc algebra of Khovanov defined by the first author. We investigate the connection of this algebra to Cautis and Kamnitzer’s recent work on link homology via coherent sheaves and suggest directions for future research.

EQUIVARIANT COLORED 𝔰𝔩(N)-HOMOLOGY FOR LINKS

In this sequel to [A colored 𝔰𝔩(N)-homology for links in S3, preprint (2009), arXiv:0907.0695v5], we construct an equivariant colored 𝔰𝔩(N)-homology for links, which generalizes both the above mentioned paper and the paper by [D. Krasner, Equivariant sl(n)-link homology, preprint (2008), arXiv:0804.3751v2]. The construction is a straightforward generalization of the paper, [A colored 𝔰𝔩(N)-homology for links in S3]. The proof of invariance is based on a simple observation which allows us to translate the proof in the above mentioned paper into the new setting. As an application, we prove that deformations over ℂ of the colored 𝔰𝔩(N)-homology are link invariants. We also construct a spectral sequence connecting the colored 𝔰𝔩(N)-homology to its deformations over ℂ, which generalizes the spectral sequence given in the papers by [B. Gornik, Note on Khovanov link cohomology, preprint (2004), arXiv:math.QA/0402266 and E. Lee, An endomorphism of the Khovanov invariant, Adv. Math.197(2) (2005) 554–586].

SO(3) homology of graphs and links

The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks are included.

The 1,2-coloured HOMFLY-PT link homology

In this paper we define the 1,2-coloured HOMFLY-PT triply graded link homology and prove that it is a link invariant. We also conjecture on how to generalize our construction for arbitrary colours.

ON CONJECTURES ABOUT POSITIVE BRAID KNOTS AND ALMOST ALTERNATING TORUS KNOTS

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

On link homology theories from extended cobordisms

This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into ℝ^3 . Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webster from Seidel–Smith construction, and the odd Khovanov homology fit into this setting. Moreover, we prove that any (strict) EQFT based on a ℤ_2 -extension of the embedded cobordism category that coincides with Khovanov homology after reducing the coefficients modulo 2 gives rise to a link invariant homology theory isomorphic to that of Khovanov.

When the theories meet: Khovanov homology as Hochschild homology of links

We show that Khovanov homology and Hochschild homology theories share a common structure. In fact they overlap: Khovanov homology of the (2,n) torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov–Rozansky sl(n) homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplication-free version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and extended here to the \mathbb{M} -reduced case, and in the case of a polygon extended to noncommutative algebras. In this framework we prove that for any unital algebra \mathcal{A} the Hochschild homology of \mathcal{A} is isomorphic to graph cohomology over \mathcal{A} of a polygon. We expect that this paper will encourage a flow of ideas in both directions between Hochschild/cyclic homology and Khovanov homology theories.

Abelian link invariants and homology

We consider the link invariants defined by the quantum Chern–Simons field theory with compact gauge group U(1) in a closed oriented 3-manifold M. The relation of the Abelian link invariants with the homology group of the complement of the links is discussed. We prove that, when M is a homology sphere or when a link—in a generic manifold M—is homologically trivial, the associated observables coincide with the observables of the sphere S3. Finally, we show that the U(1) Reshetikhin–Turaev surgery invariant of the manifold M is not a function of the homology group only, nor a function of the homotopy type of M alone.

Link Homologies and the Refined Topological Vertex

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorifying quantum group invariants associated with other representations (R_1, R_2). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.

AN ORIENTED MODEL FOR KHOVANOV HOMOLOGY

We give an alternative presentation of Khovanov homology of links. The original construction rests on the Kauffman bracket model for the Jones polynomial, and the generators for the complex are enhanced Kauffman states. Here we use an oriented sl(2) state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces. Functoriality in original Khovanov homology holds up to sign. Variants of Khovanov homology fixing functoriality were obtained by Clark–Morrison–Walker [7] and also by Caprau [6]. Our construction is similar to those variants. Here we work over integers, while the previous constructions were over gaussian integers.

Integral HOMFLY‐PT and sl(n)‐Link Homology

Using the diagrammatic calculus for Soergel bimodules, developed by Elias and Khovanov, as well as Rasmussen′s spectral sequence, we construct an integral version of HOMFLY‐PT and sl(n)‐link homology.

Sl(N)–link homology (N≥ 4) using foams and the Kapustin–Li formula

We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozansky's.

OPEN-CLOSED TQFTS EXTEND KHOVANOV HOMOLOGY FROM LINKS TO TANGLES

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant.

On the first group of the chromatic cohomology of graphs

The Hochschild homology of the algebra of truncated polynomials \({{\mathcal {A}_m=\mathbb {Z}[x]/(x^m)}}\) is closely related to the Khovanov-type homology as shown by the second author. In the present paper we utilize this in the study of the first graph cohomology group of an arbitrary graph G with v vertices. The complete description of this group is given for m = 2, 3. For the algebra \({\mathcal {A}_2}\) we relate the chromatic graph cohomology with the Khovanov homology of adequate links. We describe the chromatic cohomology over the algebra \({\mathcal {A}_3}\) using the homology of a cell complex built on the graph G. In particular we prove that \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) can be isomorphic to any finite abelian group. Moreover, we give a characterization of graphs which have torsion in cohomology \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) and construct graphs which have the same (di)chromatic polynomial but different \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) .

Khovanov homology of torus links

In this paper we show that there is a cut-off in the Khovanov homology of ( 2 k , 2 k n ) -torus links, namely that the maximal homological degree of non-zero homology groups of ( 2 k , 2 k n ) -torus links is 2 k 2 n . Furthermore, we calculate explicitly the homology group in homological degree 2 k 2 n and prove that it coincides with the center of the ring H k of crossingless matchings, introduced by M. Khovanov in [M. Khovanov, A functor-valued invariant for tangles, Algebr. Geom. Topol. 2 (2002) 665–741, arXiv:math.QA/0103190]. This gives the proof of part of a conjecture by M. Khovanov and L. Rozansky in [M. Khovanov, L. Rozansky, A homology theory for links in S 2 × S 1 , in preparation]. Also we give an explicit formula for the ranks of the homology groups of ( 3 , n ) -torus knots for every n ∈ N .

ON KHOVANOV'S COBORDISM THEORY FOR $\mathfrak{su}_3$ KNOT HOMOLOGY

We reconsider the [Formula: see text] link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of complexes of "cobordisms with seams" (actually, a "canopolis"), making it local in the sense of Bar-Natan's local [Formula: see text] theory of [2]. We show that this "seamed cobordism canopolis" decategorifies to give precisely what you had both hope for and expect: Kuperberg's [Formula: see text] spider defined in [14]. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory. Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the [Formula: see text] homology of the (2,n) torus knots.

Matrix factorizations and link homology II

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.

A geometric model for Hochschild homology of Soergel bimodules

An important step in the calculation of the triply graded link homology theory of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL(n). We present a geometric model for this Hochschild homology for any simple group G, as equivariant intersection homology of B x B-orbit closures in G. We show that, in type A these orbit closures are equivariantly formal for the conjugation T-action. We use this fact to show that in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and describe its Hilbert series, proving a conjecture of Jacob Rasmussen.

The foam and the matrix factorizationsl3link homologies are equivalent

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.

The Thurston polytope for four-stranded pretzel links

In this paper we use Heegaard Floer link homology to determine the dual Thurston polytope for pretzel links of the form P(-2r_1-1, 2q_1, -2q_2, 2r_2+1) where r_i and q_i are positive integers. We apply this result to determine the Thurston norms of spanning surfaces for the individual link components, and we explicitly construct norm-realizing examples of such surfaces.