Lipshitz--Sarkar stable homotopy type for certain planar trivalent graphs with perfect matchings

Abstract

We develop a space-level refinement of the $2$-factor homology by constructing a stable homotopy type associated to a certain family $\mathscr{G}$ of planar trivalent graphs equipped with perfect matchings. Specifically, we define a cover functor from the $2$-factor flow category $\mathscr{C}(Γ_{M})$ to the cube flow category $\mathscr{C}_{C}(n)$, where the perfect matching graph $Γ_{M}$ represents a planar trivalent graph $G$ together with a perfect matching $M$, such that $(G,M) \in \mathscr{G}$. By applying the Cohen--Jones--Segal realization to the $2$-factor flow category $\mathscr{C}(Γ_{M})$, we obtain the $2$-factor spectrum. This spectrum serves as a space-level version of the $2$-factor homology, analogous to the Lipshitz--Sarkar Khovanov spectrum for links. We show that the cohomology of the $2$-factor spectrum with $\mathbb{Z}_{2}$-coefficients is isomorphic to the $2$-factor homology, as defined by Baldridge. We prove that the stable homotopy type of the $2$-factor spectrum is an invariant of planar trivalent graphs $G$ equipped with perfect matchings $M$, whenever $(G, M) \in \mathscr{G}$. Furthermore, we show that the closed webs obtained by performing flattenings at each crossing of an oriented link diagram in the context of $\mathfrak{sl}_{3}$ link homology belong to the family $\mathscr{G}$.