Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surface-knot as a link cobordism between empty links. In this paper, we define an invariant of a surface-knot which is a generalization of the Khovanov-Jacobsson number by using Bar-Natan's theory, and prove that any $T^2$-knot has the trivial Khovanov-Jacobsson number.
We prove that a crossing change along a double point circle on a 2-knot is realized by ribbon-moves for a knotted torus obtained from the 2-knot by attaching a 1-handle. It follows that any 2-knots for which the crossing change is an unknotting operation, such as ribbon 2-knots and twist-spun knots, have trivial Khovanov-Jacobsson number.
To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for $9_{46}$; is not hindered by detecting connected sums with knotted $2$-spheres.
We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E_2-pages can be described in terms of groups arising from the action of a certain endomorphism on mod 2 Khovanov homology. Some simple consequences are discussed.
We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by D.Bar-Natan, and modified by M.Khovanov. We compute the structure of the modules for some manifolds, including Seifert fibred manifolds.
We show that over the binary field $\mathbb F_2$, the Bar-Natan perturbation of Khovanov homology splits as the direct sum of its two reduced theories, which we also prove are isomorphic. This extends Shumakovitch's analogous result for ordinary Khovanov homology, without the perturbation.
We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes \ell}\$, and admitting a direct generalization to other Lie algebras. While the construction discussed is a special case of that given in previous work of the author, this paper contains new results about the special case of \$\mathfrak{sl}_2\$ showing an explicit connection to B…
We prove a lower bound on the ribbon distance via Bar-Natan Homology. This lower bound agrees with Alishahi's lower bound on the unknotting number via Bar-Natan Homology, which furthers a pattern first observed by Sarkar: Sarkar's lower bound on the ribbon distance via Lee Homology agreed with Alishahi and Dowlin's lower bound on the unknotting number via Lee Homology. We also prove a lower bound on the ribbon distance via $α$-homology, a variant of Khovanov homology defined recently by Khovanov and Robert.
A spectral sequence is established, whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology, in Khovanov's classification.
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}…
We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant under Reidemeister moves. The terms of this chain complex 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 theo…
We show that the order of torsion homology classes in Bar-Natan deformation of Khovanov homology is a lower bound for the unknotting number. We give examples of knots that this is a better lower bound than |s(K)/2|, where s(K) is the Rasmussen s invariant defined by the Bar-Natan spectral sequence.
We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map $Φ$. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that specific families of such knots cannot be detected by Khovanov's thickness criteria. We also exhibit a sequence of prime links related by quasipositive modification whose width is increasing.
A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2 version of $F_5$ homology, in Khovanov'sclassification. Concordance invariants of knots are derived from the corresponding instanton homology groups, including a 1-parameter family of homomorphisms $f_r$ from the concordance group to the reals, having the potential to provide independent bounds on the genus and number of double points for immersed surfaces wi…
We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations have `seams' separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps…
We introduce two monoidal supercategories: the odd dotted Temperley-Lieb category $\mathcal{T\!L}_{o,\bullet}(δ)$, which is a generalization of the odd Temperley-Lieb category studied by Brundan and Ellis, and the odd annular Bar-Natan category $\mathcal{BN}_{\!o}(\mathbb{A})$, which generalizes the odd Bar-Natan category studied by Putyra. We then show there is an equivalence of categories between them if $δ=0$. We use this equivalence to better understand the action of the Lie superalgebra $\mathfrak{gl}(1|1)$ on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and…
We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu--Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the $\mathfrak{gl}_{2}$ skein lasagna module, hence is an invariant of $4$-dimensional $2$-handlebodies.
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spirit as Rasmussen's s-invariant.
This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the Jones polynomial from Khovanov homology becomes the trace of a unitary transformation on a Hilbert space associated with the Khovanov Homology.
In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams of crossing change, which we call the "genus-one morphism." It is proved that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain an invariant for singular tangles that extending sl(2) tangle homology; examples include Lee homology, Bar-Nat…
Based on the results of the second author, we define an equivariant version of Lee and Bar-Natan homology for periodic links and show that there exists an equivariant spectral sequence from the equivariant Khovanov homology to equivariant Lee homology. As a result we obtain new obstructions for a link to be periodic. These obstructions generalize previous results of Przytycki and of the second author.
This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system.
We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. We use a mapping cone construction instead of a tensor product, which allows us to deal efficiently with the more complicated sign assignments in the odd theory. The algorithm has been implemented in a computer program. We also use the algorithm to determine the odd Khovanov homology of 3-strand torus links.
We provide a unified framework for proving Reidemeister-invariance and functoriality for a wide range of link homology theories. These include Lee homology, Heegaard Floer homology of branched double covers, singular instanton homology, and \Szabo's geometric link homology theory. We follow Baldwin, Hedden, and Lobb (arXiv:1509.04691) in leveraging the relationships between these theories and Khovanov homology. We obtain stronger functoriality results by avoiding spectral sequences and instead showing that each theory factors through Bar-Natan's cobordism-theoretic link homology theory.
We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathemati…
O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskaya's class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both refinements are determined by the classical invariants of K.
This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to $\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to $\mathfrak{sl}(n)$ homology for $n \geq 2$, including the universal $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$ homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and B…
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that, for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the …
We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams with values in the homotopy category attached to the cobordism category. Motivated by Bar-Natan's categorification of the Jones polynomial, it categorifies the quantum $sl_n$ quantum invariants and is adapted to the categorification of the $sl_n$ quantum invariants by Khovanov and Rozansky using matrix factorizations. We conjecture to exist th…
We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of the third Reidemeister move. Then, to show functoriality, we modify arguments used by Clark, Morrison, and Walker to show that induced chain maps are invariant under Carter and Saito's movie moves.
We construct a bigraded (co)homology theory which depends on a parameter a, and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl(3) theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl(2) invariant (or Lee's modification of it) corresponds to a=0 (or a=1). In particular, our construction naturally resolves the sign ambiguity in the functoriality of Khovanov's sl(2) homology theory.
Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented topological quantum field theory (TQFT) to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equ…
We use the divide-and-conquer and scanning algorithms for calculating Khovanov cohomology directly on the Lee- or Bar-Natan deformations of the Khovanov complex to give an alternative way to compute Rasmussen $s$-invariants of knots. By disregarding generators away from homological degree 0 we can considerably improve the efficiency of the algorithm. With a slight modification we can also apply it to a refinement of Lipshitz-Sarkar.
We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. 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 su(2) theory of math.GT/0410495. We show that this `seamed cobordism canopolis' decategorifies to give precisely what you'd both hope for and expect: Kuperberg's su(3) spider defined in q-alg/9712003. We conjecture an answe…
For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure o…
In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called $β$-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar's invariants $ψ^\pm$. From the $β$-invariants we extract two non-negative integers which are transverse invariants (the $c$-invariants). Finally, we give several conditions which imply the non-effectiveness of the $c$-…
The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical knot cobordisms. To do so, we give an alternate formulation for the Manturov definition of Khovanov homology for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We then discuss the implications of the maps induced in the aforementioned theory to the…
We determine the algebraic structure underlying the geometric complex associated to a link in Bar-Natan's geometric formalism of Khovanov's link homology theory (n=2). We find an isomorphism of complexes which reduces the complex to one in a simpler category. This reduction enables us to specify exactly the amount of information held within the geometric complex and thus state precisely its universality properties for link homology theories. We also determine its strength as a link invariant relative to the different topological quantum field theories (TQFTs) used to create link homology. We i…
We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex, whose homotopy type modulo certain relations is a link invariant. Both the original and the odd Khovanov homology can be recovered from this construction by applying certain strict 2-functors. We …
Khovanov homology: pro-tangles, derived colimits and spectral sequences
Mathematical Science Research Center, Chongqing University of Technology（重庆理工大学数学科学研究中心）；NSF-Simons National Institute for Theory and Mathematics in Biology（NSF-西蒙斯生物理论与数学国家研究所）
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauffman bracket skein module of I-bundles over surfaces. This is accomplished by borrowing ideas from Bar-Natan's Khovanov homology theory for tangles and cobordisms and using embedded surfaces to generate the chain groups, instead of diagrams.
