Intersections of Dual $SL_3$-Webs

Abstract

We introduce a topological intersection number for an ordered pair of $\operatorname{SL}_3$-webs on a decorated surface. Using this intersection pairing between reduced $(\operatorname{SL}_3,\mathcal{A})$-webs and a collection of $(\operatorname{SL}_3,\mathcal{X})$-webs associated with the Fock--Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced $(\operatorname{SL}_3,\mathcal{A})$-webs to the tropical set $\mathcal{A}^+_{\operatorname{PGL}_3,\hat{S}}(\mathbb{Z}^t)$, as established by Douglas and Sun in \cite{DS20a, DS20b}. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fock--Goncharov duality conjecture of higher Teichm\"uller spaces for $\operatorname{SL}_3$.