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

TL;DR

This paper constructs a stable homotopy type for planar trivalent graphs with perfect matchings, providing a space-level refinement of 2-factor homology that is invariant under graph transformations.

Contribution

It introduces a new space-level invariant for graphs with perfect matchings, extending the Lipshitz--Sarkar spectrum framework to a broader class of combinatorial objects.

Findings

The 2-factor spectrum's cohomology matches 2-factor homology with Z2 coefficients.

The stable homotopy type is an invariant of the graph and perfect matching.

Webs from link diagrams are included in the family G.

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 $\mathscr{C}(\Gamma_{M})$ to the cube flow category $\mathscr{C}_{C}(n)$, where the perfect matching graph $\Gamma_{M}$ 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 $\mathscr{C}(\Gamma_{M})$, 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 $\mathscr{G}$.