sl(3) Khovanov module and the detection of the planar theta-graph

We introduce two invariants called sl(3) Khovanov module and pointed sl(3) Khovanov homology for spatial webs (bipartite trivalent graphs). Those invariants are related to Kronheimer-Mrowka’s instanton invariants $J^{\mathrm{♯}}$ and $I^{\mathrm{♯}}$ for spatial webs by two spectral sequences. As an application of the spectral sequences, we prove that sl(3) Khovanov module and pointed sl(3) Khovanov homology both detect the planar theta graph.