An odd Khovanov homotopy type (Q2173716)

Khovanov homology [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)] is a far reaching generalization of the Jones polynomial, and has led to extensive studies in the last 20 years. It was used in [\textit{J. Rasmussen}, Invent. Math. 182, No. 2, 419--447 (2010; Zbl 1211.57009)] to define the celebrated concordance invariant \(s\). In [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)] \textit{R. Lipshitz} and \textit{S. Sarkar} refined Khovanov homology to a stable homotopy type, which they then used to further refine Rasmussen's \(s\)-invariant. Motivated by attempts to connect Khovanov homology to Heegaard-Floer homology, \textit{P. S. Ozsváth} et al. [Algebr. Geom. Topol. 13, No. 3, 1465--1488 (2013; Zbl 1297.57032)] defined a variation of Khovanov homology, known as odd Khovanov homology, which is better suited for this task. Over \(\mathbb{Z}_2\)-coefficients this agrees with the original Khovanov homology, and therefore also categorifies the Jones polynomial. In this paper, the authors now define a stable homotopy type for odd Khovanov homology. Rather than using the approach of Lipshitz-Sarkar, Burnside categories as used in [\textit{T. Lawson} et al., ``Khovanov homotopy type, Burnside category, and products'', Preprint, \url{arXiv:1505.00213}] are utilized, which they further generalize to get a \(\mathbb{Z}_2\) equivariant CW-spectrum. With this technique they also get a \(\mathbb{Z}_2\)-equivariant CW-spectrum for the even theory, which they conjecture to be stable homotopy equivalent to the original Lipshitz-Sarkar construction. They even get a spectrum for the unified Khovanov homology of \textit{K. K. Putyra} and \textit{A. N. Shumakovitch} [J. Knot Theory Ramifications 25, No. 3, Article ID 1640012, 18 p. (2016; Zbl 1361.57016)]. Interestingly, the resulting action of the Steenrod algebra on odd Khovanov homology gives rise to concordance invariants that refine the \(s\)-invariant. However, there does not appear to be calculations where these differ from the homological invariants. Given the results in the even case, one should be hopeful that there do exist knots for which these invariants differ.