On SO(3) representation spaces of spatial trivalent graphs

Given a spatial trivalent graph G, we will first review some of the objects and results from Kronheimer and Mrowka’s theory about the instanton invariant J#(G). Motivated by the Tutte relation and the 4-color theorem, we will then proceed to studying decompositions of graphs into two 4-ended tangle-graphs along a 4-punctured sphere. The resulting Lagrangian Floer theory happens to be on the pillowcase. Viewing J# invariant from this angle, we will propose several modifications to the construction of representation varieties: two different reductions at two basepoints, and a passage to the equivariant theory (which corresponds to the wrapped Floer theory on the pillowcase). The advantage is that the resulting equivariant invariants are as simple as possible, and can be used to recover the initial unreduced invariants via mapping cones. Based on this we will speculate on the existence of the corresponding instanton-theoretic curve invariants in the pillowcase, and indicate an open-ended strategy for how to study the Tutte relation for J#(G). This is joint work in progress with Fan Ye.