Khovanov homotopy type. This goes via Rozansky’s definition of .

Khovanov homotopy type. 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. 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. Kauffman, Igor Mikhailovich Nikonov, and Eiji Ogasa Abstract. , 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. 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]. In this talk we will review definitions of the Jones polynomial and Khovanov homology, and give a few of their most spectacular applications. We The primary purpose of this paper is to study the Khovanov homotopy type and the annular Khovanov homotopy type of periodic links. Geom. This provides a topological refinement, whereas our work provides an algebraic one. Roughly, they turn the Khovanov cube into a functor from the cube category to the Burnside category of finite sets and correspondences. Lipshitz–Sarkar constructed an algorithm for computing the first two Steenrod squar Abstract. 24. [Lobb et al. 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. This subsection is devoted to a crucial example of flow category, which will be the fundamental building block of the Khovanov stable homotopy type. It may be regarded as a categorification of the Jones polynomial. 1, 1–75. 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]. The Bar Natan-Lee-Turner spectral sequence for fields up to characteristic 211. 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. 1261 Marco Mackaay, Marko Stošić, and Pedro Vaz, A diagrammatic categorification of the q -Schur algebra, Quantum Topol. We show that the Khovanov homotopy types of torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become stably homotopy equivalent as . 48 (2016) 327–360) and, as a corollary, that those two constructions give equivalent spectra. 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. 1017/S0143385700000183 Brent Everitt, Robert Lipshitz, Sucharit Sarkar, and Paul Turner, Khovanov homotopy types and the Dold-Thom functor. 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. 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. 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. Soc. 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. 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. ) 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. 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. 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. 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. 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. 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. 60–81. 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. This includes all links with up to 11 crossings, and [LS14b] provides a list of the stable homotopy types for all such links. In this paper we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. 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. be/33YxtArjwFAslide: https://speakerdeck. 2140/gt. 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$>1$. Burnside category. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen. The Zm -action on the Khovanov complex preserves the filtration (1. 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. The main goal of the talk will be to show that the Khovanov homotopy type can be effectively used to study periodic links, i. We construct a variant of the Khovanov homology -- the equivariant Khovanov homology -- which is adapted to the equivariant setting. 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. 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. 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)). SS was supported by NSF Grant DMS In this paper, we give a new construction of a Khovanov homotopy type. In particular, Marko Stošić proved that the homology groups stabilize as . The construc- tion of the stable homotopy type In this paper, we give a new construction of a Khovanov homotopy type. A conjecture on the 3-colored unknot. 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. Andrew Lobb, Patrick Orson, and Dirk Schütz, A Khovanov stable homotopy type for colored links, Algebr. 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. 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. 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. Geometry & Topology, Vol. That is, if a link L0 is equivariantly isotopic to L, then X (L0) is Borel homotopy equivalent to X (L). 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. 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. We also construct a Z/2 action on an even Khovanov homotopy type, with fixed point set a desuspension of X^j_o (L). ' Scholarly sources with full text pdf download. 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. 2016. Both the Jones polynomial and its categori cation, the Khovanov homology, are known to stabilize for torus links T (n; m) as m ! 1. 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. Lobb and P. 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. 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 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 analyze the homotopy type of independence complexes of circle graphs, with a focus on th se arising when the graph is bipartite. 4 (2013), no. ovanov homotopy type in Subsection 3. 1, 163–228 (2006; Zbl 1117. 57005)] to the setting of stable homotopy theory. 623 Abstract We give a new construction of a Khovanov stable homotopy type, or spectrum. In this paper, we give a new construction of a Khovanov stable homotopy type, or spectrum. 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]. 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. On a Khovanov homotopy type. So, to construct a Khovanov stable homotopy type, it suffices to construct a flow category for Khovanov homology. L/. 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. 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. 17. 2. 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. Mandell [Adv. Invariance of the Khovanov homotopy type is proved in Section 6. In 2004, Bar-Natan published [Bar04] a description of the Khovanov Bracket, [[L]] as a homotopy category over the cobordisms. (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. We define Khovanov-Lipshitz-Sarkar stable homotopy type for K to be that for K. , DOI 10. 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. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. 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. By applying Dwyer-Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link. , links which are invariant under a finite order rotation of the three-sphere. Khovanov [Duke Math. May 1, 2015 · In this paper, we give a new construction of a Khovanov homotopy type. columbia. Related research topic ideas. Indiana University Mathematics Journal, To appear. 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)). 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. 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. It is a link type invariant. Khovanov homology is a refinement of the Jones polynomial of a knot. This gives rise to a much deeper structure of invariants than the set of polynomial coefficients. The following two theorems constitute the central geometric part of the present article. sl (3)-link homology and corresponding spectral sequences. We will show that the stable homotopy type of the space is a link invariant. 2014. 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. May 1, 2015 · This decomposition allows us to compute the homotopy type of the almost-extreme Khovanov spectra of diagrams without alternating pairs. The approach by the authors is to refine the TQFT to a sophisticated enough functor that the construction of A. 16 (2016), 483-508. We investigate external group actions on homotopy coherent diagrams. We relate the Borel cohomology of X L to the equivariant Khovanov homology of L constructed by the second author. We show that this construction gives a space stably homotopy equivalent to the Khovanov spectra constructed by Lipshitz and Sarkar (J. 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. 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. "A Khovanov homotopy type"; the referee suggested adding the word "stable" to avoid confusion. e. Additional material. 4310/HHA. 3460. pdf. 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]. X. 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. In this paper, we give a new construction of a Khovanov homotopy type. Jan 14, 2021 · The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. ﻻ يوجد ملخص باللغة العربية In this paper, we give a new construction of a Khovanov homotopy type. S -invariants coming from sl (3)-link homology for fields up to characteristic 211, both knots and links. 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. arXiv:1202. 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. 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 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. This goes via Rozansky's de nition of a . Topol. 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. There are two recent constructions of Khovanov homotopy types, using different techniques and giving different results [ET14, LS14a]. 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. Abstract The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link. 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. 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. The hypercube flow category. 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. Amer. 3. 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. 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. 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. Relevant books, articles, theses on the topic 'Khovanov homotopy type. Lc/ whose cohomology recovers the c–colored Khovanov cohomology of L. It was developed in the late 1990s by Mikhail Khovanov. Abstract Khovanov homology is a bi-graded abelian group (or ring module) associated to any knot or oriented link in S3. 27 (2014) 983–1042) and Hu, Kriz and Kriz (Topology Proc. Abstract. The original publication is available here. Sep 28, 2021 · We define stable homotopy refinements of Khovanov’s arc algebras and tangle invariants. 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. (You should write a proof of that as an exercise if you don't already know it. 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. 2020. We Jan 6, 2025 · These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. iii Lay Summary Both highly visual and conceptually simple, knots are, at least at a basic level, compre- hensible to the non-mathematician. 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. 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]. 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]. X. 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. Math. The Khovanov homotopy type is a stronger invariant than Khovanov homology. 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. 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. The question whether different choices of moduli spaces lead to the same stable homotopy type is open. gl (2) foams and the Khovanov homotopy type. 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]. arXiv:1112. ABSTRACT. 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]. Jones polynomial was one of the first invariants of knots which was not geometrically defined and its precise geometric meaning is still a mystery. A more powerful invariant than the Jones polynomial, this special type of categorification has been extensively developed over the last 20 years. Jun 24, 2020 · We also construct a reduced odd Khovanov homotopy type and the unified Khovanov homotopy type. 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. 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. 19001)] can be May 1, 2015 · In this paper, we give a new construction of a Khovanov homotopy type. The homology is an invariant for oriented links up to isotopy by applying a tautological functor on the geometric complex. 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. 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. 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. 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. 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. If you know a ring structure, write a paper and wow a lot of people. 3, 359–426 (2000; Zbl 0960. 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. 4. 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. The action of Steenrod algebra on the cohomology of XL gives an extra structure of the periodic link. 17 (2017), 1261-1281. 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. Krushkal, V. 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. Intersection homology of linkage spaces in odd dimensional Euclidean space, Algebr. There are two recent con-structions of Khovanov homotopy types, using different techniques and giving different results [3, 6]. These computations are motivated by the question whether the constructed Khovanov homotopy type actually contains more information about an oriented link than Khovanov homology. 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. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link. In particular, the results imply that the equivariant Khovanov homotopy types defined by [BPS21] and [SZ18] are equivariantly stably homotopy equivalent. 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. Andrew Lobb will deliver a sequence of lectures in the topic, and his lecture series will be complemented by some lectures of other participants. X (L) is a suspension spectrum of a CW complex with cellular cochain complex satis-fying C (X (L)) ' KC (L). We show the author's result obtained with Kau man, and the author's result obtained with Kau man and Nikonov. 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. The construction of X^j(L) is combinatorial and explicit. 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. 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. Finally in this lecture we define Khovanov homology and discuss some properties. 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. The Khovanov Homotopy Type of In nite Torus Links. 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. 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. This produces a Khovanov homotopy type whose reduced homology is Khovanov homology. 04769) are Oxford Academic Loading Jun 4, 2021 · We have an elementary introduction to Khovanov-Lipshitz-Sarkarstable 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. Jul 23, 2018 · In this paper we study Khovanov homology of periodic links. These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. 205, No. Lipshitz and Sarkar strengthened this inviariant in [LSa], constructing a homotopy type whose cohomology was the Khovanov homology of a link. To extend this argument to the Khovanov homotopy type, there are two di culties. 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 rest of this section is devoted to showing that the cobordism maps on Khovanov homology commute with stable cohomo ogy operations (like Steenrod squares). 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 develop a space-level formulation of Khovanov skein homology by constructing a stable homotopy type for annular links. The Blanchet link homology theory is an oriented model of Kho-vanov homology, functorial over the integers with respect to link cobordisms. (in press). 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. L/ is a suspension spectrum of a CW complex with cellular cochain complex satisfying C . 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. A difference persists in that [BPS21] identified the Borel cohomology of their spectrum with equivariant Khovanov homology as defined by Politarczyk in [Pol19]. 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. This section is devoted ing terminology to discuss these sequences of Kauffman states. 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 will discuss recent progress on understanding and applying L-Sarkar's \Khovanov homotopy type". 1. 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. We ask an open question in the author's paper written with Kauffman and Nikonov. We relate the Borel cohomology of XL to the equivariant Khovanov homology of L constructed by the second author. May 2, 2024 · Our main result has topological, combinatorial and computational flavor and is connecting four fundamental conjectures: Conjecture 1. Thus, Khovanov homology inherits Steenrod operations. Section 5. 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. 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. 4 will be constructed as homotopy colimits of the Khovanov spectra for link diagrams as defined in [7]. 1856. 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. edu), 141 Cabell Drive, Kerchof Hall, PO Box 400137, Charlottesville, VA 229044137. 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 the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying Such links in S1 × D2 are also called m-periodic. L/ of a link L, which we shall refer to as the Lipshitz–Sarkar–Khovanov (abbreviated as L-S-K) spectrum of L. 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. Semantic Scholar extracted view of "An odd Khovanov homotopy type" by Sucharit Sarkar et al. 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. 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. 17 (2017), no. 2017. Khovanov homology actually has another grading, making it more precisely a categorification of the representation theory of a quantum group. Where Admission Tyler Lawson, Robert Lipshitz* (lipshitz@math. (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. 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). This produces a Khovanov homotopy type whose able homotopy type is a knot invariant. The action of Steenrod algebra on the cohomology of X L gives an extra structure of the periodic link. 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. 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. As part of this program, the actions of the standard generators of sl2 are lifted to maps of spectra. 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]. We prove that Jones Polynomial is equal to a suitable Euler characteristic Remark 1. specifically, of its Khovanov homology. In particular, the results imply that the equivariant Khovanov homotopy types defined by Borodzik, Politarczyk, Silvero (arXiv:1807. be/IGrBsXkazVM3/3: https://youtu. edu) and Sucharit Sarkar. MR 3623688, DOI 10. v18. Elmendorf and M. 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. We set up foundations of representation theory over S, the stable sphere, which is the \initial ring" of stable homotopy theory. 101, No. 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. 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. This is used to prove an equivalence between realizations of equivariant cubical flow categories and external actions on Burnside functors. 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 Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. 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. J. be/QZF8eWULfNA2/3: https://youtu. The homology has also geometric descriptions by introducing the genus generating operations. 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. 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). 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. Lipshitz-Sarkar constructed an algorithm for computing the first two Steenrod squares. Abstract Khovanov homology is a combinatorially-de ned invariant of knots and links, with various generalizations to tangles. 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. It is the first meaningful Khovanov-Lipshitz-Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. com/taketosano/khovanov-homolog Geometry & Topology Volume 28, issue 4 (2024) Bibliography Burnside category. 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. 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. 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. 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. 30 Mar 20, 2021 · 1/3: https://youtu. 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. 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]. Our goal is to sample some recent applications of Khovanov-type theories to smooth low-dimensional topology. 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. 1), hence it descends to a Zm -action on the annular Khovanov chain complex. 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. This goes via Rozansky's de nition of a VYACHESLAV KRUSHKAL AND PAUL WEDRICH ABSTRACT. 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. Recently, Lawson, Lipshitz, and Sarkar generalized Khovanov homology to a spectrum-valued Khovanov homotopy type, from which the Khovanov ho-mology can be recovered. 2, 1261–1281. 08795) and Stoffregen, Zhang (arXiv:1810. We consider the problem of lifting this action to the stable homotopy re nement of the annular homology. 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. 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]. 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. Syllabus: We will study Khovanov homology, which is a very modern invariant of knots and a categorification of the famous Jones polynomial. 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 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. a9 Brent Everitt and Paul Turner, The homotopy theory of Khovanov homology. 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. 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. Like the construction of Khovanov homology, our construction of the Khovanov low category is entirely combinatorial. 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. 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. 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. 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]. 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. 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. This goes via Rozansky’s definition of Our Khovanov homotopy type will be defined more complicated families of Kauffman states. We show that the Abstract. 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]. The primary purpose of this paper is to study the Khovanov homotopy type and the annular Khovanov homotopy type of periodic links. We propose to study the S-type (stable homotopy type) of Thorn complexes in the framework of the Atiyah-Adams J-Theory. By bringing together the various ideas and constructions, we hope to facilitate new applications. 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. In fact, we show an algorithm to determine extreme Khovanov homotopy type of those braids in polynomial time. We establish various properties such as fixed point constructions and cofibration sequences. 18 (2014) 17-30. 2140/agt. "Errata to 'A cylindrical reformulation of Heegaard Floer homology'". 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. 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. L// ' KC . By lifting this functor to the Burnside category, one can construct a CW complex whose reduced cellular chain complex agrees with the Khovanov complex. 30--10. They were used by Lipshitz-Sarkar in [LS14a] to produce a stable homotopy type link invariant XKh(L) for links L S3. Orson, Algebr. 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. 6 The Khovanov spectra of Theorem 1. 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. It is the first meaningful Abstract. Abstract We give a new construction of a Khovanov stable homotopy type, or spectrum. In particular, KH detects the unknot which at the moment of writing is still unknown Abstract. First Announcement This weekend workshop is organized to understand recent developements in Khovanov homology, with a special attention to Khovanov homotopy type. 1. VYACHESLAV KRUSHKAL AND PAUL WEDRICH ABSTRACT. 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. 4. 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. Remark 1. The refinement is to a stable homotopy type (finite CW-spectrum), and spectra do not have cohomology rings: stabilization destroys the cup product. 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. 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. , & Wedrich, P. This goes via Rozansky's definition of a Program of the workshopThursday, January 18, 2018 9. n2. In particular, Marko Stosic proved that the homology groups stabilize as . 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. 14. Mar 15, 2016 · We study a Khovanov type homology close to the original Khovanov homology theory from Frobenius system. In particular, its We have an elementary introduction to Khovanov-Lipshitz-Sarkar stable homotopy type. 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. bdq2 xza sqh8e9f hvpjt 4usv5 txv6gelq faw k64 ct0o dsta2