A colored Khovanov spectrum and its tail for B-adequate links
In recent work, \textit{R. Lipshitz} and \textit{S. Sarkar} [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)] introduced a stable homotopy type for Khovanov cohomology, that is, a suspension spectrum whose cohomology recovers Khovanov cohomology. In this paper, the author extends their construction to \(\mathfrak{sl}_2(\mathbb{C})\)-colored links. Indeed, the author obtains a spectrum for every link diagram involving a finite number of Jones-Wenzl projectors with the correct cohomology. This gives a space level version of the work in [\textit{B. Cooper} and \textit{V. Krushkal}, Quantum Topol. 3, No. 2, 139--180 (2012; Zbl 1362.57015)], and [\textit{L. Rozansky}, Quantum Topol. 5, No. 4, 541--579 (2014; Zbl 1326.57032)]. Following Rozansky, the author shows that for \(n\)-colored \(B\)-adequate links the Khovanov spectra stabilize as \(n\) goes to infinity. For the \(n\)-colored unknot the resulting stable homotopy type can be viewed as a limit \(\mathcal{X}(T(n,\infty))\) of the Lipshitz-Sarkar stable homotopy type of the \(n\)-braided torus knot. Sending \(n\) to infinity then gives rise to a stable limit \(\mathcal{X}(T(\infty,\infty))\).
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