Towards an $\mathfrak{sl}_2$ action on the annular Khovanov spectrum
Rostislav Akhmechet, Vyacheslav Krushkal, Michael Willis
备注：
55 pages, many figures, comments welcome! This new version contains an additional corollary in the introduction and a new section on link cobordism maps, together with a handful of minor changes
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 of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.
In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an application of this spectral sequence, we prove that the annular Khovanov homology detects the unlink in the thickened annulus (assuming all the components are null-homologous). Another application is a new proof of Grigsby and Ni's result that tangle Khovanov homology distingui…
We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with $U(1) \times U(1)$-equivariant cohomology of $\mathbb{CP}^1$. Motivated by the relationship between the Temperley-Lieb algebra and annular Khovanov homology, we also introduce an equivariant analogue of the Temperley-Lieb algebra.
We define a third grading on Khovanov homology, which is an invariant of annular links but changes by $\pm 1$ under stabilization. We illustrate the use of our computer implementation, and give some example calculations.
We construct a braid conjugacy class invariant $κ$ by refining Plamenevskaya's transverse element $ψ$ in Khovanov homology via the annular grading. While $κ$ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using $κ$ we construct an obstruction to negative destabilization (stronger than $ψ$) and a solution to the word problem in braid groups. Also, $κ$ is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete exampl…
Let $\widetilde{K}$ be a two-periodic braid and let $K$ be its quotient. In this paper we show there is a spectral sequence from the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of $\widetilde{K}$ to the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of $K$.
We define an annular version of odd Khovanov homology and prove that it carries an action of the Lie superalgebra $\mathfrak{gl}(1|1)$ which is preserved under annular Reidemeister moves.
We exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste-Przytycki and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture. We also extend the Batson-Seed link splittin…
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…
This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.
Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n. One therefore obtains a "knotted" Schur-Weyl representation that agrees with classical sl_2 Schur-Weyl duality when K is the Seifert-framed unknot.
We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for bra…
Let $L$ be a link in a thickened annulus. Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $sl_2(\wedge)$, the exterior current algebra of the Lie algebra $sl_2$. In this paper, we upgrade this result to the setting of $L_\infty$-algebras and modules. That is, we show that $sl_2(\wedge)$ is an $L_\infty$-algebra and that the annular Khovanov homology of $L$ is an $L_\infty$-module over $sl_2(\wedge)$. Up to $L_\infty$-quasi-isomorphism, this structure is invariant under Reidemeister moves. Finally, we include explicit formulas to compute the …
We give some new link detection results for link Floer homology, Khovanov homology and annular Khovanov homology. The links we detect arise via different closure operations on $3$-braids. Examples of our results include that link Floer homology detects the Mazur link, that annular Khovanov homology detects the Mazur pattern, and that Khovanov homology detects L6a2 and L9n15. The Mazur pattern detection result depends on a new bound on the rank of the annular Khovanov homology of certain links.
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.
For a 2-periodic link $\tilde L$ in the thickened annulus and its quotient link $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}].$ This spectral sequence splits along quantum and $sl_2$ weight space gradings, proving a rank inequality $rank\ AKh^{j,k}(L) \leq rank\ AKh^{2j-k,k} (\tilde L)$ for every pair of quantum and $sl_2$ weight space gradings $(j,k)$. We also present a few decategorified consequences and discuss partial results toward a sim…
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…
Lawrence Roberts, extending the work of Ozsvath-Szabo, showed how to associate to a link, L, in the complement of a fixed unknot, B, in S^3, a spectral sequence from the Khovanov homology of a link in a thickened annulus to the knot Floer homology of the preimage of B inside the double-branched cover of L. In a previous paper, we extended Ozsvath-Szabo's spectral sequence in a different direction, constructing for each knot K in S^3 and each positive integer n, a spectral sequence from Khovanov's categorification of the reduced, n-colored Jones polynomial to the sutured Floer homology of a red…
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.
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…
We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov…
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 define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora, but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces, which recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.
In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.
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…
Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, to a graded algebra and a graded bimodule over it, they associate a quantum Hochschild homology of the algebra with coefficients in the bimodule, and use this to obtain a deformation of the annular Khovanov homology of a link. A spectral refinement of the resulting invariant was recently given by Akhmechet-Krushkal-Willis. In this short note we observe that quantum Hochschild homology is a composition of two familiar operations, and give a short proof that it gives an invariant of …
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 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.
We define a wall-crossing morphism for Khovanov-Rozansky homology; that is, a map between the KR homology of knots related by a crossing change. Using this map, we extend KR homology to an invariant of singular knots categorifying the Vasilliev derivative of the HOMFLY polynomial, and of $\mathfrak{sl}_n$ quantum invariants.
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 define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids.
We construct a supercategory that can be seen as a skew version of (thickened) KLR algebras for the type $A$ quiver. We use our supercategory to construct homological invariants of tangles and show that for every link our invariant gives a link homology theory supercategorifying the Jones polynomial. Our homology is distinct from even Khovanov homology and we present evidence supporting the conjecture that it is isomorphic to odd Khovanov homology. We also show that cyclotomic quotients of our supercategory give supercategorifications of irreducible finite-dimensional representations of $\math…
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}.
In this paper we use Kuperberg's $\mathfrak{sl}_3$-webs and Khovanov's $\mathfrak{sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{sl}_3$-web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$-skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus}_0(\mathcal{W}^S)_{\mathbb{Q}(q)}$, to show that its center is…
We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus …
