Stranding $\mathfrak{sl}_n$ webs

Abstract: Webs are a kind of planar, directed, edge-labeled graph that encode invariant vectors for quantum representations of $\mathfrak{sl}_n$. The theory of webs developed organically for $\mathfrak{sl}_2$, where they are also known as noncrossing matchings and the Temperley-Lieb algebra, before being formalized by Kuperberg for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ as the morphisms in a diagrammatic categorification of quantum representations called the spider category. Various models extend webs to $n \geq 4$. Only Cautis-Kamnitzer-Morrison prove a full set of relations for their webs, though Fontaine's webs are better adapted to computations, more graph-theoretically natural, and directly generalize webs for $n=2$ and $n=3$. This paper formalizes the theory of Fontaine's webs, proving the existence of a deep and powerful global structure on these webs called strandings. We do three key things: 1) give a state-sum formula to construct ($U_q(\mathfrak{sl}_n)$-invariant) web vectors from the orientation of strandings on Fontaine's webs; 2) list and prove a complete set of relations, connecting strandings to the local data of binary labelings that are well-established in the literature; and 3) provide applications and examples of how strandings facilitate computations.