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

TL;DR

This paper introduces new invariants for spatial webs, specifically the sl(3) Khovanov module and pointed sl(3) Khovanov homology, which are related to instanton invariants and can detect the planar theta graph.

Contribution

The paper defines two novel invariants for spatial webs and establishes their relation to existing instanton invariants via spectral sequences, demonstrating their ability to detect the planar theta graph.

Findings

sl(3) Khovanov module and pointed sl(3) Khovanov homology are introduced.

These invariants are related to Kronheimer-Mrowka's instanton invariants through spectral sequences.

Both invariants can detect the planar theta graph.

Abstract

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 $J^\sharp$ and $I^\sharp$ 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.