A Khovanov stable homotopy type for colored links (Q518844)

In the present paper, the authors provide a well defined homotopy type associated to every colored link \(L\) whose cohomology recovers the colored Khovanov homology of \(L\). Categorification aims at improving polynomial invariants of links by interpreting them as graded Euler characteristics of some bigraded cohomology groups associated to each link. Khovanov homology [\textit{M. Khovanov}, Commun. Algebra 29, No. 11, 5033--5052 (2001; Zbl 1018.16015)] pioneered the field by categorifying the Jones polynomial, and it developed then in several directions. One of these directions is the categorifcation of colored Jones polynomials, which has been adressed, in particular, by \textit{L. Rozansky} [Fundam. Math. 225, 305--326 (2014; Zbl 1336.57025)] and \textit{B. Cooper} and \textit{V. Krushkal} [Quantum Topol. 3, No. 2, 139--180 (2012; Zbl 1362.57015)]. The two constructions are equivalent, but the former one defines its invariant for a \(c\)-colored link \(L\) as the stable limit of the Khovanov homologies of the \(c\)-cable of \(L\) with an increasing number of full twists inserted on each component, whereas the latter one provides a more explicit categorification of the Jones-Wenzl projectors, given in terms of \textit{D. Bar-Natan}'s reformulation of Khovanov theory [Geom. Topol. 9, 1443--1499 (2005; Zbl 1084.57011)]. After categorification, the next step is spacification, that is the interpretation of (bigraded) cohomological invariants as the cohomology groups of some well defined (split) homotopy types associated to each link; this has been initiated by \textit{R. Lipshitz} and \textit{S. Sarkar}'s work on Khovanov homology in [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)]. The present paper combines Lipshitz-Sarkar's construction with Rozansky's categorifaction to provide a spacification for colored Jones polynomials. In the 2-colored case, the authors discuss also an alternative but equivalent approach, based on the more explicit Cooper-Krushkal construction, and use it to provide enlightening computations. The main results of the paper are proved in Section 2. In Section 2.2, the authors draw on Rozansky's computation for the \((rn,n)\)-torus braid to show that the Khovanov homologies of cable links with an increasing number of full twists on each component do stabilize in each degree; they can then use the Lipshitz-Sarkar procedure to spacify it. In Section 2.3, they present a frame for what a lift of Cooper-Krushkal projectors to framed flow categories -- to which some CW-complex is naturally associated -- should be; and they prove in Section 2.4 that any lift satisfying this frame would lead to the same homotopy type as one with the Rozansky-based construction. Such a lift is explicitly given for the 2-colored case in Section 3; and this leads to computations for the 2-colored unknot and trefoil, and for the \((2,1)\)-colored Hopf link in Sections 4.1--4.3. The paper ends with a conjecture and a discussion on the 3-colored unknot case. Note that the paper assumes some familiarity with [Cooper and Krushkal, loc. cit.] and [Lipshitz and Sarkar, loc. cit.].