Khovanov homotopy type. VYACHESLAV KRUSHKAL AND PAUL WEDRICH ABSTRACT.
Khovanov homotopy type. It is the first meaningful Khovanov-Lipshitz-Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. So, to construct a Khovanov stable homotopy type, it suffices to construct a flow category for Khovanov homology. a9 Brent Everitt and Paul Turner, The homotopy theory of Khovanov homology. Jul 23, 2018 · Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. Topological and algebraic exposition are sprinkled In Section 3, we brie y discuss Lipshitz and Sarkar's explicit formulas for computing the rst two Steenrod squares on Khovanov homology in terms of the generators of the Khovanov chain groups [LS12]. This section is devoted ing terminology to discuss these sequences of Kauffman states. The construction of the stable homotopy type relies on the signed Burnside category May 1, 2015 · In this paper, we give a new construction of a Khovanov homotopy type. SS was supported by NSF Grant DMS In this paper, we give a new construction of a Khovanov homotopy type. 3460. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying The Khovanov homotopy type is a space level refinement of Khovanov homology in troduced by Lipshitz and Sarkar. 14. We develop a space-level formulation of Khovanov skein homology by constructing a stable homotopy type for annular links. The construction of X^j(L) is combinatorial and explicit. A Khovanov homotopy type is a way of associating a (stable) space to each link L so that the classical invariants of the space yield the Khovanov homology of L. The cells of the resulting CW complex are in one-to-one correspondence with the generators In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. Another idea of categorification came Jan 7, 2019 · The other limitation which should be mentioned concerns any knot stable homotopy type, including the s l 2 -Khovanov homotopy type. This is used to prove an equivalence between realizations of equivariant cubical flow categories and external actions on Burnside functors. Khovanov homology is a refinement of the Jones polynomial of a knot. We will discuss recent progress on understanding and applying L-Sarkar's \Khovanov homotopy type". If you know a ring structure, write a paper and wow a lot of people. The question whether different choices of moduli spaces lead to the same stable homotopy type is open. 2. This family includes the Khovanov-Lipshitz-Sarkar stable homotopy type for the homotopical Khovanov homology of links in higher genus surfaces (see the 1117-55-273 Michael S Willis* (msw3ka@virginia. Moreover, we com-pute (real) extreme Khovanov homology of a 4-strand pretzel knot using ch Rhea Palak Bakshi Department of Mathematics, University of California, Santa In this talk we show some advances on the conjecture, showing that computing the result holds when considering extreme Khovanov homology of closed braids of at most 4 strands. be/QZF8eWULfNA2/3: https://youtu. 2140/agt. We formulate a stable homotopy refinement of the Blanchet An intermediate possibility would be to replace the Khovanov homology by an abstract space or simplicial object whose generalized homotopy type was an invariant of the knot or link. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. 2014. Khovanov homology actually has another grading, making it more precisely a categorification of the representation theory of a quantum group. The Khovanov-Frobenius Algebra as a BV Algebra This section establishes the Batalin-Vilkovisky (BV) algebra structure on the equivariant Khovanov-Frobenius algebra from [13]. We also construct a Z/2 action on an even Since Khovanov [Kho99] introduced his homology theory for links in 1999, there has been a lot of progress in categori cation of knot polynomials, and investigation on knot homology theories in general. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying In [6], Robert Lipshitz and Sucharit Sarkar defined a stable homotopy type in-variant X. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying In this paper, we give a new construction of a Khovanov homotopy type. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors Another refinement of Kho-vanov homology is the Khovanov homotopy type, a spectrum XKh(L) whose coho-mology is Khovanov homology [18]. Khovanov homology lifts the Jones polynomial one level higher, and discovers surprising connections between Nov 26, 2022 · Khovanov homology is not a ring. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. The hypercube flow category. "A Khovanov homotopy type"; the referee suggested adding the word "stable" to avoid confusion. It is the first meaningful Abstract. Andrew Lobb will deliver a sequence of lectures in the topic, and his lecture series will be complemented by some lectures of other participants. Another consequence of our construction is Apr 9, 2019 · The Lipshitz–Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. X. We analyze the homotopy type of independence complexes of circle graphs, with a focus on th se arising when the graph is bipartite. Therefore we focus our attention on the group JR (X) which is defined to be the group of orthogonal sphere bundles over X modulo In this paper, we give a new construction of a Khovanov homotopy type. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying Such links in S1 × D2 are also called m-periodic. The construction of the stable homotopy type relies on the signed Burnside category The primary purpose of this paper is to study the Khovanov homotopy type and the annular Khovanov homotopy type of periodic links. 17 (2017), 1261-1281. 1. Khovanov Homology & Khovanov Homotopy type (3/3) - さのたけと / すうがく徒のつどい@online (2021-03-20) Taketo Sano 1. 2017. Another consequence of We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. I After the categorification, people invented further ehnancements of knot invariants, say, spectrification and Khovanov homotopy where with a knot diagram we associate some topological space (spectrum) whose homotopy type is invariant under Reidemeister moves. This goes via Rozansky's de nition of a VYACHESLAV KRUSHKAL AND PAUL WEDRICH ABSTRACT. A stable homotopy re nement of Khovanov homology endowed it with extra structure, such as an action by the Steenrod algebra [LS14c], which was then used to construct a family of additional s-type concordance invariants [LS14b], as well as to show that Khovanov homotopy type is a strictly stronger invariant than Khovanov homology [See]. 02K subscribers Subscribed Dec 16, 2015 · Former titles: "A Khovanov homotopy type or two"; using an idea of Oleg Viro we can now show the two variants agree. Jun 24, 2020 · We also construct a reduced odd Khovanov homotopy type and the unified Khovanov homotopy type. For an arbitrary link L ⊂ S3, Sarkar-Scaduto-Stoffregen construct a family Xl(L), l ≥ 0, of spaces, giving a family of spatial refinements of even and odd Khovanov homology. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen. We show that the construction behaves well with respect to disjoint unions, connected Historical Introduction II Khovanov homology (KH) offers a nontrivial generalization of the Jones polynomial (and the Kauffman bracket polynomial) of links in R3. Both the Jones polynomial and its categori cation, the Khovanov homology, are known to stabilize for torus links T (n; m) as m ! 1. By using the… Expand 8 [PDF] Feb 3, 2016 · We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. columbia. That is, if a link L0 is equivariantly isotopic to L, then X (L0) is Borel homotopy equivalent to X (L). In Feb 11, 2016 · We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. L/ is a suspension spectrum of a CW complex with cellular cochain complex satisfying C . The homology has also geometric descriptions by introducing the genus generating operations. arXiv:1112. Orson, Algebr. ) What you do get is an action of the Steenrod algebra on Khovanov We would like to use the Khovanov homotopy type to study smoothly embedded surfaces in R4. 18 (2014) 17-30. Mentioning: 5 - Given an m-periodic link L ⊂ S 3 , we show that the Khovanov spectrum XL constructed by Lipshitz and Sarkar admits a homology group action. We show that A stable homotopy re nement of Khovanov homology endowed it with extra structure, such as an action by the Steenrod algebra [LS14c], which was then used to construct a family of additional s-type concordance invariants [LS14b], as well as to show that Khovanov homotopy type is a strictly stronger invariant than Khovanov homology [See]. 3. A difference persists in that [BPS21] identified the Borel cohomology of their spectrum with equivariant Khovanov homology as defined by Politarczyk in [Pol19]. Additional material. The starting point is an innocent-looking function \ (f \colon \mathbb {R} \to \mathbb {R}\) with only two critical points, at 0 and 1, which are respectively a minimum and a maximum. The construction categorifies the Jones polynomial, by taking the Euler characteristic of the Khovanov Jan 21, 2025 · between the Khovanov homologies associated to the two knots. To extend this argument to the Khovanov homotopy type, there are two di culties. Expand 10 PDF Khovanov homotopy type, periodic links and localizations Maciej BorodzikWojciech PolitarczykMarithania Silvero Mathematics Mathematische Annalen 2021 TLDR By applying the Dwyer–Wilkerson theorem, Khovanov homology of the quotient link is expressed in terms of equivariant Khovanova homological of the original link. pdf. 17 (2017), no. The primary purpose of this paper is to study the Khovanov homotopy type and the annular Khovanov homotopy type of periodic links. Khovanov stable homotopy type and higher representation theory -Meng GUO 郭萌, UIUC(2024-12-05） Khovanov homology, a categorification of the Jones polynomial, assigns a graded chain complex to links and tangles, with its Euler characteristic recovering the polynomial. 4 will be constructed as homotopy colimits of the Khovanov spectra for link diagrams as defined in [7]. 101, No. Roughly, they turn the Khovanov cube into a functor from the cube category to the Burnside category of finite sets and correspondences. The Zm -action on the Khovanov complex preserves the filtration (1. 2016. This gives rise to a much deeper structure of invariants than the set of polynomial coefficients. We extend Lipshitz and Sarkar’s definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. S -invariants coming from sl (3)-link homology for fields up to characteristic 211, both knots and links. Like Khovanov homology, the Khovanov homotopy types tri ially satisfy an unoriented skein triangle; this is explained in Sect Aug 9, 2016 · It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. This goes via Rozansky’s definition of Our Khovanov homotopy type will be defined more complicated families of Kauffman states. These computations are motivated by the question whether the constructed Khovanov homotopy type actually contains more information about an oriented link than Khovanov homology. Amer. Lipshitz and Sarkar strengthened this inviariant in [LSa], constructing a homotopy type whose cohomology was the Khovanov homology of a link. This includes all links with up to 11 crossings, and [LS14b] provides a list of the stable homotopy types for all such links. ABSTRACT. We give a computation of Sq2on these spaces, determining the stable homotopy type of Xl(K) for all l and all knots K up to 11 crossings. J. It was developed in the late 1990s by Mikhail Khovanov. In the case of n 𝑛 n -colored B-adequate links, we show a stabilization of the homotopy types as the coloring n → ∞ → 𝑛 n\rightarrow\infty, generalizing the tail behavior of the Mar 31, 2021 · The Khovanov chain complex is a categorification of the Jones polynomial and is built using a functor from Kauffman’s cube of resolutions to Abelian groups. In particular, its homotopy type, if not contractible, would We extend Lipshitz and Sarkar’s definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Nov 9, 2022 · An invariant of link cobordisms from Khovanov homology Khovanov homotopy type, Burnside category and products Planar algebras and the decategorification of bordered Khovanov homology Fixing the functoriality of Khovanov homology: a simple approach Dualizability in low-dimensional higher category theory Aug 9, 2021 · In this article, we focus on the impact of the most influential homology theory arising from quantum invariants: Khovanov homology [22]. Oct 10, 2024 · This subsection is devoted to a crucial example of flow category, which will be the fundamental building block of the Khovanov stable homotopy type. 1 (a) Computing Khovanov homology of a closed braid with fixed number of strands has polynomial time complexity with respect to the number of crossings. On a Khovanov homotopy type. They were written following a lecture series given by Sucharit Sarkar at the Renyi Institute during a special semester on "Singularities and low-dimensional topology", organised by the Erdos Center. We show that the construction behaves well with respect to disjoint unions, connected Part 2: Khovanov homology and homotopy State sums and the Jones polynomial The Khovanov cube Applications of Khovanov homology Structure of Khovanov homotopy type Sep 18, 2025 · Khovanov-Lipshitz-Sarkar stable homotopy type for the homotop-ical Khovanov chain complex ([11, 16]) of K. Jul 23, 2018 · View a PDF of the paper titled Khovanov homotopy type, periodic links and localizations, by Maciej Borodzik and 2 other authors Sep 19, 2025 · A proof would explain the topological phenomena of the homotopy type as consequences of the intrinsic algebraic properties of the Khovanov-Sano complex. Relevant books, articles, theses on the topic 'Khovanov homotopy type. In 2004, Bar-Natan published [Bar04] a description of the Khovanov Bracket, [[L]] as a homotopy category over the cobordisms. Given an assignment c (called a coloring) of positive… Expand 3 [PDF] The structure of the Khovanov homology of torus links has been extensively studied. 205, No. Syllabus: We will study Khovanov homology, which is a very modern invariant of knots and a categorification of the famous Jones polynomial. and , On the Khovanov and knot Floer homologies of quasi-alternating links, Proceedings of Gökova Geometry-Topology Conference 2007, Gökova Geometry/Topology Conference (GGT), Gökova, 2008, pp. Geometry & Topology, Vol. 4. In this paper, we give a new construction of a Khovanov stable homotopy type, or spectrum. Invariance of the Khovanov homotopy type is proved in Section 6. Indiana University Mathematics Journal, To appear. We prove that Jones Polynomial is equal to a suitable Euler characteristic Remark 1. We give a new construction of a Khovanov stable homotopy type, or spectrum. In [10] and [11] one of the authors showed that the spectra of closures of infinite Mar 31, 2021 · Venue: MS teams (team code hiq1jfr) The Khovanov chain complex is a categorification of the Jones polynomial and is built using a functor from Kauffman’s cube of resolutions to Abelian groups. be/IGrBsXkazVM3/3: https://youtu. This subsection is devoted to a crucial example of flow category, which will be the fundamental building block of the Khovanov stable homotopy type. As an application, we construct a Khovanov slk-stable homotopy type with a large prime We then show that the natural $\mathbb {Z}/p\mathbb {Z}$ action on a $p$-periodic link $L$ induces such an action on Lipshitz and Sarkar's Khovanov functor $F_ {Kh} (L): \CC \rightarrow \mathscr {B}$ which makes the Khovanov homotopy type $\mathcal {X} (L)$ into an equivariant knot invariant. In particular, Marko Stošić proved that the homology groups stabilize as . There are two recent con-structions of Khovanov homotopy types, using different techniques and giving different results [3, 6]. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored Khovanov cohomology of L. The original publication is available Home University Archives University of Oregon Administration Office of the Vice President for Research and Innovation Theses and Dissertations Equivariant Khovanov Homotopy Type and Periodic Links A Khovanov homotopy type is a way of associating a (stable) space to each link L so that the classical invariants of the space yield the Khovanov homology of L. Abstract. The construc- tion of the stable homotopy type ArXiv MathSciNet Link We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. This produces a Khovanov homotopy type whose reduced homology is Khovanov homology. 57005)] to the setting of stable homotopy theory. Largely independently, inspired by their homotopy-theoretic investigations of topolog-ical and conformal field theories, Hu, Kriz and Kriz gave another construction of a Khovanov stable homotopy type with the same basic properties [22]. We relate the Borel cohomology of XL to the equivariant Khovanov homology of L constructed by the second author. We also construct a Z/2 action on an even Khovanov homotopy type, with fixed point set a desuspension of X^j_o (L). Abstract We give a new construction of a Khovanov stable homotopy type, or spectrum. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col (L_c) whose cohomology recovers the c-colored Khovanov cohomology of L. As part of this program, the actions of the standard generators of sl2 are lifted to maps of spectra. Make the set of all Khovanov-Lipshitz-Sarkar stable homotopy types for all λ and for a fixed choice of the right and the left. We develop a new … Aug 29, 2021 · Khovanov Homotopy Type Khovanov homology は, Jones polynomial を categorify するものであるが, その定義 (構成) は chain complex を用いたもので, 代数的トポロジーの視点からは, とても古臭く感じる。 代数的トポロジーにおけるホモロジー は, 公理で規定されるものであり, より具体的には spectrum を用いて表される Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. e. Determining the homotopy type of the geometric realization of Kho-vanov homology (Khovanov spectra) of a closed braid with fixed num-ber of strands has polynomial time complexity with respect to the number of crossings. Apr 26, 2018 · This is a space-level construction of Khovanov homology whose stable homotopy type is a well-defined invariant of the isotopy type of the link. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying Abstract We give a new construction of a Khovanov stable homotopy type, or spectrum. In particular, its We have an elementary introduction to Khovanov-Lipshitz-Sarkar stable homotopy type. In fact, we show an algorithm to determine extreme Khovanov homotopy type of those braids in polynomial time. In this paper, we give a new construction of a Khovanov homotopy type. (b) Determining the homotopy type of the geometric realization of Khovanov homology (Khovanov spectra) of a The primary purpose of this paper is to study the Khovanov homotopy type and the annular Khovanov homotopy type of periodic links. Major topics include the Jones polynomial, Khovanov homology, Bar-Natan's cobordism category, applications of Khovanov homology, some spectral sequences, Khovanov stable homotopy type, and skein lasagna modules. The Bar Natan-Lee-Turner spectral sequence for fields up to characteristic 211. To prove this, we simply need to associate a map of Khovanov spectra to each cobor-dism and verify that the induced map on cohomology agrees with t This course is an introduction to Khovanov homology of knots and the Heegaard Floer homology of knots and 3-manifolds, with an emphasis on their topological applications. Jan 19, 2018 · For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o(L) whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. Mar 21, 2012 · Then, we discuss, in this framework, two recent results (independent of each other) on refinements of Khovanov homology: our refinement into a module over the connective k-theory spectrum and a stronger result by Lipshitz and Sarkar refining Khovanov homology into a stable homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov spectra constructed in [LS14a] and [HKK16] and, as a corollary, that those two constructions give equivalent spectra. com/taketosano/khovanov-homolog Geometry & Topology Volume 28, issue 4 (2024) Bibliography Burnside category. Given an assignment c (called a coloring) of a positive integer to each component of a link L, we define a stable homotopy type Xcol. We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the homotopical Khovanov homology of links in thickened surfaces indexed by moduli space systems. Andrew Lobb, Patrick Orson, and Dirk Schütz, A Khovanov stable homotopy type for colored links, Algebr. This produces a Khovanov homotopy type whose Mar 31, 2021 · Venue: MS teams (team code hiq1jfr) The Khovanov chain complex is a categorification of the Jones polynomial and is built using a functor from Kauffman’s cube of resolutions to Abelian groups. Abstract Khovanov homology is a bi-graded abelian group (or ring module) associated to any knot or oriented link in S3. We define Khovanov-Lipshitz-Sarkar stable homotopy type for K to be that for K. We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. Following Khovanov and Bar-Natan, as a step towards this goal, in this paper we construct an extension of the Khovanov stable homotopy type to tangles. Soc. Recently, Lawson, Lipshitz, and Sarkar generalized Khovanov homology to a spectrum-valued Khovanov homotopy type, from which the Khovanov homology can be recovere Jan 31, 2024 · We investigate group actions on homotopy coherent diagrams. Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus$>1$ are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. Jan 11, 2024 · These notes provide an introduction to the stable homotopy types in Khovanov theory (due to Lipshitz-Sarkar) and in knot Floer theory (due to Manolescu-Sarkar). Major topics include the Jones polynomial, Khovanov homology, Bar-Natan’s cobordism category, applications of Khovanov homology, some spectral sequences, Khovanov stable homotopy type, and skein lasagna modules. X. Jul 23, 2018 · In this paper we study Khovanov homology of periodic links. 24. We Jul 17, 2020 · Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus> 1 are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus> 1. We ask an open question in the author's paper written with Kauffman and Nikonov. 2020. Dec 16, 2011 · We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. We show that the Khovanov homotopy types of torus links, … ArXiv MathSciNet Link We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). Khovanov homotopy type, periodic links and localizations Maciej BorodzikWojciech PolitarczykMarithania Silvero Mathematics Mathematische Annalen 2021 TLDR By applying the Dwyer–Wilkerson theorem, Khovanov homology of the quotient link is expressed in terms of equivariant Khovanova homological of the original link. This thesis is primarily a ground-up survey of the Khovanov homotopy type; beginning with the Jones Given an m -periodic link L ⊂ S 3 , we show that the Khovanov spectrum X L constructed by Lipshitz and Sarkar admits a group action. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying Jun 5, 2021 · We show easy examples of explicit construction of Khovanov-Lipshitz-Sarkar stable homotopy type. gl (2) foams and the Khovanov homotopy type. We also prove that the Steenrod squares Sq2 0, Sq2 June 29, 2018 - This talk was part of the 2018 RTG mini-conference Low-dimensional topology and its interactions with symplectic geometry MR 1756976, DOI 10. We will show that the stable homotopy type of the space is a link invariant. The stable homotopy type of the 0-framed 3-colored unknot Xj col(U3) was partially computed by Willis [11], who showed that it was not a wedge of Moore spaces and so, in some sense, more interesting than just the colored Khovanov cohomology. Finally, we also provide an alternative, simpler stabilization in the case of the Nov 9, 2015 · The structure of the Khovanov homology of torus links has been extensively studied. edu) and Sucharit Sarkar. 4. Finally in this lecture we define Khovanov homology and discuss some properties. Spacifying other knot homology theories Derived Representation Theory and Stable Homotopy Categorification of sl_k - Hu, Kriz and Somberg An sl_n stable homotopy type for matched diagrams - Jones, Lobb and Schuetz A Khovanov stable homotopy type for colored links - Lobb, Orson and Schuetz An odd Khovanov homotopy type - Sarkar, Scaduto, Stoffregen Both approaches furnish localization results relating the Khovanov homotopy type of a periodic link to the annular Khovanov homotopy type of its quotient, resulting in periodicity criteria. 1), hence it descends to a Zm -action on the annular Khovanov chain complex. In [CS], Seed proved the Steenrod square alone is a stronger invariant than Mar 20, 2021 · Khovanov Homology & Khovanov Homotopy type (1/3) - さのたけと / すうがく徒のつどい@online (2021-03-20) Taketo Sano 1. This goes via Rozansky’s definition of Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). In particular, Marko Stosic proved that the homology groups stabilize as . The refinement is to a stable homotopy type (finite CW-spectrum), and spectra do not have cohomology rings: stabilization destroys the cup product. The homology is an invariant for oriented links up to isotopy by applying a tautological functor on the geometric complex. We Jan 6, 2025 · These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. v18. X (L) is a suspension spectrum of a CW complex with cellular cochain complex satis-fying C (X (L)) ' KC (L). In particular, KH detects the unknot which at the moment of writing is still unknown Abstract. We relate the Borel cohomology of X L to the equivariant Khovanov homology of L constructed by the second author. Introduction Framed ow categories were introduced by Cohen-Jones-Segal in [CJS95] as a way potentially to re ne Floer homological invariants to space-level invariants. Jul 17, 2020 · Abstract and Figures We define the Khovanov-Lipshitz-Sarkar homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened higher genus surfaces. This article is a Our Khovanov-Lipshitz-Sarkar homotopy type and our Steenrod square are stronger than the homotopical Khovanov homology of links in thickened higher genus surfaces. Krushkal, V. The approach by the authors is to refine the TQFT to a sophisticated enough functor that the construction of A. specifically, of its Khovanov homology. fundamental diference between Alexander polynomial and Jones (and HOMFLYPT) polynomial is that Alexander polynomial can be computed in polynomial time while finding Jones (and Aug 15, 2025 · The second Steenrod Square for the Lipshitz-Sarkar stable Khovanov homotopy type, and an odd version. 2140/gt. We also construct a \mathbb {Z}/2 action on an even Khovanov homotopy type, with fixed point set a desuspension of subscript \mathcal {X}_ {o}^ {j} (L). We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. First Announcement This weekend workshop is organized to understand recent developements in Khovanov homology, with a special attention to Khovanov homotopy type. We show that this con-struction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. Sep 19, 2021 · This family includes the Khovanov-Lipshitz-Sarkar stable homotopy type for the homotopical Khovanov homology of links in higher genus surfaces (see the content of the paper for the definition). 08795) and Stoffregen, Zhang (arXiv:1810. Elmendorf and M. Lecture 1-2 (1/23, 1/30): Sections 1-4 of [LS1] (Lecture 1 Notes) Framed flow categories and their realizations as (cubical) CW-complexes The cube flow category Lecture 3-4 (2/6, 2/13): Section 5 of [LS1] The Khovanov flow category Ladybug matching and the moduli spaces associated to decorated resolution configuration diagrams The definition of the Khovanov homotopy type Lecture 5 (2/20 A stable homotopy re nement of Khovanov homology endowed it with extra structure, such as an action by the Steenrod algebra [LS14c], which was then used to construct a family of additional s-type concordance invariants [LS14b], as well as to show that Khovanov homotopy type is a strictly stronger invariant than Khovanov homology [See]. Witten 2011 argued, following indications in Gukov, Schwarz & Vafa 2005, that this 4d TQFT is related to the worldvolume theory of the image in type IIB string theory of D3-branes ending on NS5-branes in a type II supergravity background of the form ℝ 9 × S 1 with the circle transverse to both kinds of branes, under one S-duality Khovanov homology is a combinatorially-defined invariant of knots and links, with various generalizations to tangles. In particular, the results imply that the equivariant Khovanov homotopy types defined by [BPS21] and [SZ18] are equivariantly stably homotopy equivalent. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying Transverse invariant as Khovanov skein spectrum at its extreme Alexander grading Nilangshu Bhattacharyya, Adithyan Pandikkadan Abstract. The original publication is available here. We construct a variant of the Khovanov homology -- the equivariant Khovanov homology -- which is adapted to the equivariant setting. Abstract Khovanov homology is a combinatorially-de ned invariant of knots and links, with various generalizations to tangles. We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. May 1, 2015 · In this paper, we give a new construction of a Khovanov homotopy type. They then apply the Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a desuspension of the even Khovanov homotopy type. Expand 10 PDF Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. The construction of the stable homotopy type relies on the signed Burnside category approach of So, to construct a Khovanov stable homotopy type, it suffices to construct a flow category for Khovanov homology. Kauffman, Igor Mikhailovich Nikonov, and Eiji Ogasa Abstract. Mandell [Adv. Indeed, the resulting Steenrod algebra action on Khovanov homology is nontrivial [LS14c, See12], leading to a spectrum-level refinement of Rasmussen’s s-invariant [LS14b]. Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus> 1 are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus> 1. We show the author's result obtained with Kau man, and the author's result obtained with Kau man and Nikonov ArXiv MathSciNet Link We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. The following two theorems constitute the central geometric part of the present article. Intersection homology of linkage spaces in odd dimensional Euclidean space, Algebr. Burnside category. This goes via Rozansky's definition of a Program of the workshopThursday, January 18, 2018 9. Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a May 1, 2015 · In this paper, we give a new construction of a Khovanov homotopy type. In [Wil16, Wil], one of the authors showed that the homotopy types of closures of in nite The Lipshitz-Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. In particular, we treat S-Lie algebras and their representa-tions, characters, gln(S)-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. 4310/HHA. 2, 1261–1281. This produces a Khovanov homotopy type whose able homotopy type is a knot invariant. This goes via Rozansky’s definition of We extend Lipshitz and Sarkar’s definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. L// ' KC . Our goal is to sample some recent applications of Khovanov-type theories to smooth low-dimensional topology. Hence, the spectrum-level invariants Khovanov homotopy type, Burnside category and products Tyler Lawson, Robert Lipshitz and Sucharit Sarkar Geometry & Topology 24 (2020) 623–745 DOI: 10. Lobb and P. In this talk we will review definitions of the Jones polynomial and Khovanov homology, and give a few of their most spectacular applications. Up to a finite formal desuspension, the spectra of [7] are suspension spectra of CW complexes, and thus their homotopy colimits are also suspension spectra of CW complexes (the “suspension Aug 21, 2016 · We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Jan 19, 2018 · Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a desuspension of the even Khovanov homotopy type. Remark 1. , 2017] and, independently, [Willis, ] proved that the Khovanov homotopy type stabilizes under adding twists, and used this to extend it to a colored Khovanov stable homotopy type; fu Apr 11, 2025 · A Khovanov stable homotopy type for colored links, with A. Section 5. Our Khovanov flow category has one object for each generator of the standard Khovanov complex (which is reviewed in Section 2), and the grading is the homological grading on Khovanov homology. In particular, the results imply that the equivariant Khovanov homotopy types defined by Borodzik, Politarczyk, Silvero (arXiv:1807. It was proven in [GMS] that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. 60–81. The action of Steenrod algebra on the cohomology of XL gives an extra structure of the periodic link. Abstract We define a Khovanov homotopy type for 𝔰 𝔩 2 (ℂ) 𝔰 subscript 𝔩 2 ℂ \mathfrak {sl}_ {2} (\mathbb {C}) colored links and quantum spin networks and derive some of its basic properties. arXiv:1202. By applying Dwyer-Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link. L/ of a link L, which we shall refer to as the Lipshitz–Sarkar–Khovanov (abbreviated as L-S-K) spectrum of L. [Lobb et al. Related research topic ideas. VYACHESLAV KRUSHKAL AND PAUL WEDRICH ABSTRACT. We then show that the natural Z=pZ action on a p-periodic link L induces such an action on Lipshitz and Sarkar's Khovanov functor FKh(L) : 2n ! which makes the Khovanov homotopy type X (L) into an equivariant knot B invariant. Recently, Lawson, Lipshitz, and Sarkar generalized Khovanov homology to a spectrum-valued Khovanov homotopy type, from which the Khovanov ho-mology can be recovered. (Received August 29, 2014)1 Jan 14, 2021 · The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra sl2, which is natural with respect to annular link cobordisms. L/. In the case of n –colored B –adequate links, we show a stabilization of the spectra as the coloring n → ∞ , generalizing the tail behavior of the colored Jones polynomial. 1856. ﻻ يوجد ملخص باللغة العربية In this paper, we give a new construction of a Khovanov homotopy type. In recent work, Robert Lipshitz and Sucharit Sarkar constructed the Khovanov The paper under review was motivated by the desire to refine the TQFT of M. Abstract The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We provide an explicit bound on values of beyond which the stabilization begins This stable homotopy tye induces a Steenrod square on Khovanov homology, and a result by Baues [Bau95] shows that this is enough to completely determine the Khovanov stable homotopy type of relatively simple links. For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o (L) whose cohomology recovers Ozsvath-Rasmussen-Szabos odd Khovanov homology, H_i (X^j_o (L)) = Kh^ {i,j}_o (L), following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. Thus, Khovanov homology inherits Steenrod operations. Mar 15, 2016 · We study a Khovanov type homology close to the original Khovanov homology theory from Frobenius system. 1261 Marco Mackaay, Marko Stošić, and Pedro Vaz, A diagrammatic categorification of the q -Schur algebra, Quantum Topol. 04769) are Oxford Academic Loading Jun 4, 2021 · We have an elementary introduction to Khovanov-Lipshitz-Sarkarstable homotopy type. Lipshitz-Sarkar constructed an algorithm for computing the first two Steenrod squares. We extend Lipshitz-Sarkar's de nition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. 4 (2013), no. 30 Mar 20, 2021 · 1/3: https://youtu. The Blanchet link homology theory is an oriented model of Kho-vanov homology, functorial over the integers with respect to link cobordisms. (You should write a proof of that as an exercise if you don't already know it. 05K subscribers Subscribed Sep 6, 2024 · In 2010 Seidel and Smith showed there is a spectral sequence relating the symplectic Khovanov homology of a two-periodic knot to the symplectic Khovanov homology of its quotient; in contrast, in 2018 Stoffregen and Zhang used the Khovanov homotopy type to show that there is a spectral sequence from the combinatorial Khovanov homology of a two In this lecture we begin with an aside on Frobenius algebras and topological quantum field theories and after this discuss the homotopy invariance properties of the Khovanov complex con-centrating on the first Reidemeister move. Lipshitz–Sarkar constructed an algorithm for computing the first two Steenrod squar Abstract. We investigate external group actions on homotopy coherent diagrams. 48 (2016) 327–360) and, as a corollary, that those two constructions give equivalent spectra. This provides a topological refinement, whereas our work provides an algebraic one. As stable Nov 12, 2024 · Meng Guo (UIUC): Khovanov stable homotopy type Event Type Seminar/Symposium Sponsor UIUC math department Location Altgeld 241 Date Nov 12, 2024 11:00 am Views 135 Originating Calendar Topology Seminar Khovanov homology assigns a graded chain complex of vector spaces to a link or a tangle. 1017/S0143385700000183 Brent Everitt, Robert Lipshitz, Sucharit Sarkar, and Paul Turner, Khovanov homotopy types and the Dold-Thom functor. We show the author's result obtained with Kau man, and the author's result obtained with Kau man and Nikonov. Where Admission Tyler Lawson, Robert Lipshitz* (lipshitz@math. 27 (2014) 983–1042) and Hu, Kriz and Kriz (Topology Proc. Sep 28, 2021 · We define stable homotopy refinements of Khovanov’s arc algebras and tangle invariants. These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. There are two recent constructions of Khovanov homotopy types, using different techniques and giving different results [ET14, LS14a]. Given an assignment c (called a coloring) of a positive integer to each component of a link L, we define a stable homotopy type Xcol(Lc) whose cohomology recovers the c–colored Khovanov cohomology of L. Papers related to the Lipshitz-Sarkar Khovanov stable homotopy type: I'm collecting a list of papers written about the Khovanov and Knot Floer stable homotopy types in order to get a sense of what ideas have been explored already, and who the explorers are. Jones polynomial was one of the first invariants of knots which was not geometrically defined and its precise geometric meaning is still a mystery. By lifting this functor to the Burnside category, one can construct a CW complex whose reduced cellular chain complex agrees with the Khovanov complex. This goes via Rozansky's definition of a We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link. We show that this construction gives a space stably homotopy equivalent to the Khovanov spectra constructed by Lipshitz and Sarkar (J. It is a link type invariant. After the categorification, people invented further enhancements of knot invariants, say, spectrification and Khovanov homotopy where with a knot diagram we associate some topological space (spectrum) whose homotopy type is invariant under Reidemeister moves[18]. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus > 1 are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus > 1. Lc/ whose cohomology recovers the c–colored Khovanov cohomology of L. , DOI 10. The starting point is an innocent-looking function : R → R with only two critical points, at 0 and 1, which are respectively a minimum and a maximum. In this paper we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. 623 Abstract We give a new construction of a Khovanov stable homotopy type, or spectrum. The Khovanov Homotopy Type of In nite Torus Links. Our Khovanov flow category has one object for each generator of the standard Khovanov complex (which is reviewed in Section 2), and the grading is the homological grading on Khovanov homology. Item is Freigegeben einblenden: alle ausblenden: alle A stable homotopy re nement of Khovanov homology endowed it with extra structure, such as an action by the Steenrod algebra [LS14c], which was then used to construct a family of additional s-type concordance invariants [LS14b], as well as to show that Khovanov homotopy type is a strictly stronger invariant than Khovanov homology [See]. 1. May 1, 2015 · This decomposition allows us to compute the homotopy type of the almost-extreme Khovanov spectra of diagrams without alternating pairs. edu), 141 Cabell Drive, Kerchof Hall, PO Box 400137, Charlottesville, VA 229044137. They were used by Lipshitz-Sarkar in [LS14a] to produce a stable homotopy type link invariant XKh(L) for links L S3. By bringing together the various ideas and constructions, we hope to facilitate new applications. iii Lay Summary Both highly visual and conceptually simple, knots are, at least at a basic level, compre- hensible to the non-mathematician. Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a We give a new construction of a Khovanov stable homotopy type, or spectrum. 6 The Khovanov spectra of Theorem 1. Feb 3, 2025 · Abstract and Figures It is an outstanding open question whether Jones polynomial (respectively, Khovanov homology, Khovanov-Lipshitz-Sarkar homotopy type) can be extended to all manifolds. sl (3)-link homology and corresponding spectral sequences. Furthermore, the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a desuspension of In a different direction, the Khovanov homotopy type admits a number of extensions. be/33YxtArjwFAslide: https://speakerdeck. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying 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 $\mathbb {Z}/2\mathbb {Z}$… Expand We define a Khovanov spectrum for s l 2 ( ℂ ) –colored links and quantum spin networks and derive some of its basic properties. n2. We consider the problem of lifting this action to the stable homotopy re nement of the annular homology. The action of Steenrod algebra on the cohomology of X L gives an extra structure of the periodic link. The rest of this section is devoted to showing that the cobordism maps on Khovanov homology commute with stable cohomo ogy operations (like Steenrod squares). , & Wedrich, P. We show that the Khovanov homotopy types of torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become stably homotopy equivalent as . We set up foundations of representation theory over S, the stable sphere, which is the \initial ring" of stable homotopy theory. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying Mar 15, 2012 · We will see how a cube of resolutions produces a framed flow category for the Khovanov chain complex, and how the framed flow category produces a space whose reduced cohomology is the Khovanov homology. We establish various properties such as fixed point constructions and cofibration sequences. This goes via Rozansky’s definition of The extent to which the Khovanov homotopy type is able to detect mutation is an open ques- tion, and the Khovanov homotopy type for tangles seems to be particularly well-suited for investigating this question. 2747 Oct 1, 2014 · In [13], Lipshitz and Sarkar defined the Khovanov-Lipshitz-Sarkar stable homotopy type for links in S 3 , and proved that the cohomology group of the Khovanov-Lipshitz-Sarkar stable homotopy type Abstract. This goes via Rozansky's definition of a Furthermore, the odd Khovanov homotopy type carries a ℤ 2 \mathbb {Z}/2 action whose fixed point set is a desuspension of the even Khovanov homotopy type. The rst is that we have not veri ed that the maps associated to cobordisms in 3 Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus> 1 are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus> 1. In [LS14], Robert Lipshitz and Sucharit Sarkar de ned a homotopy type invariant X (L) of a link L, which we shall refer to as the Lipshitz-Sarkar-Khovanov (abbreviated as L-S-K) homotopy type of L. Read A Khovanov stable homotopy typeLet α be a real vector bundle over a finite CW complex X and let T (α;X) be its associated Thorn complex. ovanov homotopy type in Subsection 3. 30--10. (in press). MR 3623688, DOI 10. Jan 14, 2021 · The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. 16 (2016), 483-508. 17. Topol. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying For each link L⊂S3 and every quantum grading j, we construct a stable homotopy type Xoj(L) whose cohomology recovers Ozsváth-Rasmussen-Szabó's odd Khovanov homology, H˜i(Xoj(L))=Khoi,j(L), following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. "Errata to 'A cylindrical reformulation of Heegaard Floer homology'". We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying Abstract. 19001)] can be May 1, 2015 · In this paper, we give a new construction of a Khovanov homotopy type. Like the construction of Khovanov homology, our construction of the Khovanov low category is entirely combinatorial. We show that the Abstract. Feb 20, 2024 · Their Khovanov homotopy type is then the homotopy colimit of this complex; it is not, in general, a sum of Moore spaces, and it contains more information than Khovanov homology [LS14c]. Download scientific diagram | Independence complexes associated with these graphs are contractible from publication: Homotopy type of circle graph complexes motivated by extreme Khovanov homology Louis H. Semantic Scholar extracted view of "An odd Khovanov homotopy type" by Sucharit Sarkar et al. This goes via Rozansky's de nition of a . We propose to study the S-type (stable homotopy type) of Thorn complexes in the framework of the Atiyah-Adams J-Theory. 1, 163–228 (2006; Zbl 1117. Geom. We The primary purpose of this paper is to study the Khovanov homotopy type and the annular Khovanov homotopy type of periodic links. Given an assignment c (called a coloring) of positive integer to each component of a link L, we de ne a stable homotopy type Xcol(Lc) whose cohomology recovers the c-colored Khovanov cohomology of L. Khovanov [Duke Math. ' Scholarly sources with full text pdf download. May 2, 2024 · Our main result has topological, combinatorial and computational flavor and is connecting four fundamental conjectures: Conjecture 1. DISSERTATION ABSTRACT Je rey Musyt Doctor of Philosophy Department of Mathematics June 2019 Title: Equivariant Khovanov Homotopy Type and Periodic Links In this dissertation, we give two equivalent de nitions for a group Gacting on a strictly-unitary-lax-2-functor D : 2n!B from the cube category to the Burnside category. 3, 359–426 (2000; Zbl 0960. The construc- tion of the stable homotopy type In this paper, we give a new construction of a Khovanov homotopy type. It may be regarded as a categorification of the Jones polynomial. , links which are invariant under a finite order rotation of the three-sphere. We then show that the natural Z=pZ action on a p-periodic link Linduces A stable homotopy re nement of Khovanov homology endowed it with extra structure, such as an action by the Steenrod algebra [LS14c], which was then used to construct a family of additional s-type concordance invariants [LS14b], as well as to show that Khovanov homotopy type is a strictly stronger invariant than Khovanov homology [See]. The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. The main goal of the talk will be to show that the Khovanov homotopy type can be effectively used to study periodic links, i. A conjecture on the 3-colored unknot. Math. 1, 1–75. Khovanov homotopy type is a way of associating a (stable) space to each link L so that the classical invariants of the space yield the Khovanov homology of L. The Khovanov homotopy type is a stronger invariant than Khovanov homology. Dec 16, 2011 · View a PDF of the paper titled A Khovanov stable homotopy type, by Robert Lipshitz and Sucharit Sarkar Apr 22, 2014 · Using the Khovanov homotopy type, we produce a family of generalizations of the s-invariant [LSa]; each of them is a slice genus bound, and we show that at least one of them is a stronger bound. A more powerful invariant than the Jones polynomial, this special type of categorification has been extensively developed over the last 20 years. The cohomology of the spectrum is the Khovanov cohomology of the link L and so the spectrum can be XKh(L Such links in S 1 × D 2 are also called m-periodic. 58 pw2bg khm wpbyei qo1oz5s w6djq jia2 zyw0ka ipe v8l