Webs and multiwebs for the symplectic group

We define $2n$-multiwebs on planar graphs and discuss their relation with $\mathrm{S}\mathrm{p}(2n)$-webs. On a planar graph with a symplectic local system we define a matrix whose Pfaffian is the sum of traces of $2n$-multiwebs. As application we generalize Kasteleyn’s theorem from dimer covers to $2n$-multiweb covers of planar graphs with $U(n)$ gauge group. For $\mathrm{S}\mathrm{p}(4)$ we relate Kuperberg’s ``tetravalent vertex’’ to the determinant, and classify reduced $4$-webs on some simple surfaces: the annulus, torus, and pair of pants. We likewise define, for $\mathrm{S}\mathrm{p}(2n)$ and $q=1$, a $2n$-valent vertex corresponding to the determinant, and classify reduced $2n$-webs on an annulus.