The categorification of the Kauffman bracket skein module of \(\mathbb{R}P^3\) (Q2872007)

In 2004, Asaeda, Przytycki and Sikora [\textit{M. M. Asaeda} et al., Algebr. Geom. Topol. 4, 1177-1210 (2004; Zbl 1070.57008)] constructed a categorification of the Kauffman bracket skein module of I-bundles over surfaces by generalizing Khovanov's categorification. However, in the case where the base surface is \(\mathbb{R}P^{2}\) (and the coefficient ring is not \(\mathbb{F}_{2}\)), their construction fails due to the strange behavior called ``the \(1\to 1\) bifurcations'' of links projected to \(\mathbb{R}P^{2}\). In the paper under review, the author categorifies the Kauffman bracket skein module of the twisted I-bundle over \(\mathbb{R}P^{2}\), that is \(\mathbb{R}P^{3}\setminus {*}\). The main idea is based on utilizing the wedge product instead of the tensor product when we construct chain complexes. We can deal with the problem of the \(1\to 1\) bifurcations by controlling the signs. The idea is inspired by Manturov's work, who defined Khovanov homology for virtual links with arbitrary coefficients.NEWLINENEWLINEIn Section~\(2\), the author explains the definition of the Kauffman bracket skein module of \(\mathbb{R}P^{3}\) in a careful manner.NEWLINENEWLINEIn Sections~\(3\), \(4\) and \(5\), we see the definition of the chain groups and the differentials of the categorification.NEWLINENEWLINESection~\(6\) is devoted to the proof of the invariance.NEWLINENEWLINEWhen we extend Khovanov homology to a link homology for links in general 3-manifolds or link diagrams on general surfaces, we often encounter the problem of the \(1\to 1\) bifurcations. Although we can often avoid the problem by restricting the coefficient ring to \(\mathbb{F}_{2}\), we lose a lot of information. The reviewer thinks that the author's and Manturov's ideas are beneficial to manage the problem of \(1\to 1\) bifurcations.