Khovanov Homology Research Articles

Recently, Sarkar–Scaduto–Stoffregen constructed a stable homotopy type for odd Khovanov homology, hence obtaining an action of the Steenrod algebra on Khovanov homology with Z/2Z coefficients. Motivated by their construction we propose a way to compute the second Steenrod square. Our construction is not unique, but we can show it to be a link invariant which gives rise to a refinement of the Rasmussen s-invariant with Z/2Z coefficients. We expect it to be related to the second Steenrod square arising from the Sarkar–Scaduto–Stoffregen construction.

On geometric realizations of the extreme Khovanov homology of pretzel links

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in S^{3} that do not become isotopic when their interiors are pushed into B^{4} . In particular, we identify examples where the surfaces are not even topologically isotopic in B^{4} , examples that are topologically but not smoothly isotopic, and examples of infinite families of surfaces that are distinct only up to isotopy rel. boundary. Our main proofs distinguish surfaces using the cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.

Building on the work by Alishahi–Dowlin, we extract a new knot invariant \lambda \geq 0 from universal Khovanov homology. While \lambda is a lower bound for the unknotting number, in fact more is true: \lambda is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that, for all n\geq 0 , there exists a knot K with \lambda(K) = n . Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.

Let L be a link in a thickened annulus. In Grigsby et al. (2018), Grigsby–Licata–Wehrli showed that the annular Khovanov homology of L is equipped with an action of \mathfrak{sl}_{2}(\wedge) , the exterior current algebra of the Lie algebra \mathfrak{sl}_{2} . In this paper, we upgrade this result to the setting of L_{\infty} -algebras and modules. That is, we show that \mathfrak{sl}_{2}(\wedge) is an L_{\infty} -algebra and that the annular Khovanov homology of L is an L_{\infty} -module over \mathfrak{sl}_{2}(\wedge) . Up to L_{\infty} -quasi-isomorphism, this structure is invariant under Reidemeister moves. Finally, we include explicit formulas to compute the higher L_{\infty} -operations.

Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology.

Asaeda-Przytycki-Sikora, Manturov, and Gabrovšek extended Khovanov homology to links in R P 3 \mathbb {RP}^3 . We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen’s s s -invariant in this setting. We show that the s s -invariant gives constraints on the genera of link cobordisms in the cylinder I × R P 3 I \times \mathbb {RP}^3 . As an application, we give examples of freely 2 2 -periodic knots in S 3 S^3 that are concordant but not standardly equivariantly concordant.

We study the instanton Floer homology for links in R P 3 \mathbb {RP}^3 and prove that the second page of Kronheimer–Mrowka’s spectral sequence is isomorphic to the Khovanov homology of the mirror link. As an application, we prove that Khovanov homology detects the unknot and the standard R P 1 \mathbb {RP}^1 in R P 3 \mathbb {RP}^3 .

We show the n n colored Jones polynomials of a highly twisted link approach the Kauffman bracket of an n n colored skein element. This is in the sense that the corresponding categorifications of the colored Jones polynomials approach the categorification of the Kauffman bracket of the skein element in a direct limit, as the number of full twists of each twist region tends toward infinity, proving a quantum version of Thurston’s hyperbolic Dehn surgery theorem implicit in Rozansky’s work, and giving a categorical version of a result by Champanerkar-Kofman [Algebr. Geom. Topol. 5 (2005), pp. 1–22]. In view of the volume conjecture, we compute the asymptotic growth rate of the Kauffman bracket of the limiting skein element at a root of unity and relate it to the volumes of regular ideal octahedra that arise naturally from the evaluation of the colored Jones polynomials of the link.

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.

Abstract We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots and that HOMFLY homology detects infinitely many knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that $0$ -surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R Conjecture.

A pattern for torsion in Khovanov homology

A Link Invariant Related to Khovanov Homology and Knot Floer Homology

We interpret Manolescu–Neithalath’s cabled Khovanov homology formula for computing Morrison–Walker–Wedrich’s \mathrm{KhR}_{2} skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over Bar-Natan’s cobordism category. Using categorified projectors, we compute the \mathrm{KhR}_{2} skein lasagna modules of (manifold, boundary link) pairs (S^{2} \times B^{2}, \tilde{\beta}) , where \tilde{\beta} is a geometrically essential boundary link, identifying a relationship between the lasagna module and the Rozansky projector appearing in the Rozansky–Willis invariant for nullhomologous links in S^{2} \times S^{1} . As an application, we show that the \mathrm{KhR}_{2} skein lasagna module of S^{2} \times S^{2} is trivial, confirming a conjecture of Manolescu.

We use the methods of Khovanov homology to associate a type DA structure, as defined by Lipshitz, Ozsváth and Thurston, to any even tangle diagram in a compact planar surface with boundary, with certain combinatorial data on the boundary. The homotopy type of this DA structure is an invariant of the tangle represented by the diagram. We consider the compositional properties of these DA structures when gluing boundaries in a manner similar to that in the study of planar algebras, and relate these structures to the authors previous work on tangles in discs and to the computation of Khovanov homology.

Khovanov Homology for Links in Thickened Multipunctured Disks

Z=pZ-action on .S 3 ; Q L/ which preserves Q L and whose fixed-point set is an unknot z U disjoint from Q L. A particular application of our techniques is the following: periodic link , for a prime p, with quotient link L. Let Kh. Q LI F p / denote the Khovanov homology of Q L, with coefficients in F p , the field of p elements. Let AKh.LI F p / denote the annular Khovanov homology of L, viewed in the complement of U D z U =Z p . Let Kh o . Q LI F p / and AKh o .LI F p / denote the odd Khovanov homology and annular Khovanov homology, respectively. Then dim Kh. Q LI F p / dim AKh.LI F p / and dim Kh o . Q LI F p / dim AKh o .LI F p /: In Section 5, we develop some machinery for homotopy colimits for homotopy-coherent diagrams with an external action. We do not pursue the greatest level of generality here; indeed, a more satisfactory treatment would be to essentially generalize the bulk of Vogt [56] to this situation; see also work of Dotto and Moi [20] . The main results are Proposition 5.4 and Lemma 5.6, while the main application to realizations of Burnside functors is Proposition 5.23. In fact, including Proposition 5.23 substantially increases the preliminaries we need, but is not needed in order to show that the Khovanov spaces of p-periodic links admit a Z p -action. Instead, Proposition 5.23 is only needed to show that the resulting Z p -action is well defined. In Section 6, we show that KH and KHO have external actions under suitable circumstances, and find the fixed-point functors. This involves a reasonably detailed study of the relationship of resolution configurations in a periodic link with those in its quotient. It is somewhat interesting that the case of odd Khovanov homology here is substantially more involved than the even case. We conclude the introduction with a few remarks. First, in sections dealing with homotopy-coherent diagrams, we work with diagrams in K-spaces for a group K, although for all of our applications K will always be Z 2 or trivial. We include the more general case because it is no more complicated, and also on account of a conjecture of [50] . To explain this conjecture, recall that there is an infinite family of Khovanov spaces X n .L/ of a link L for n 2 Z 0 , where the n th space has cellular chain complex equal to the even (resp. odd) Khovanov chain complex if n is even (resp. odd). The conjecture of [50] is that there should be stable homotopy equivalences .1.7/ X n .L/ ' X nC2 .L/: An attractive method of proving this conjecture would be the construction of a further Burnside functor KH Z W .2 n / op ! B Z recovering KHO.L/ by taking Z ! Z 2 . If such a functor could be constructed, our techniques would apply immediately to its realizations. Note that even if (1.7) holds, Theorem 1.3 is not entirely boring for n 2. Indeed, the statement (1.7) requires a choice of homotopy equivalence, and we expect that the natural family of homotopies realizing this equivalence (constructed from the putative KH Z ) is not contractible. That is, there may be no preferred homotopy equivalence X n ! X nC2 .

Abstract The goal of this paper is to set up the general framework of higher‐dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.

In this paper, we introduce two monoidal supercategories: the odd dotted Temperley–Lieb category [Formula: see text], which is a generalization of the odd Temperley–Lieb category studied by Brundan and Ellis in [Monoidal Supercategories, Commun. Math. Phys. 351 (2017) 1045–1089], and the odd annular Bar-Natan category [Formula: see text], which generalizes the odd Bar-Natan category studied by Putyra in [A [Formula: see text]-category of chronological cobordisms and odd Khovanov homology, Banach Center Publ. 103 (2014) 291–355]. We then show there is an equivalence of categories between them if [Formula: see text]. We use this equivalence to better understand the action of the Lie superalgebra [Formula: see text] on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and the second author in [An action of [Formula: see text] on odd annular Khovanov homology, Math. Res. Lett. 27(3) (2020) 711–742].

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname {HFT}$ and the Khovanov invariant $\widetilde {\operatorname {Kh}}$ that were developed by the authors in previous works.