On webs in quantum type $C$

We study webs in quantum type $C$, focusing on the rank three case. We define a linear pivotal category $\mathbf{W}\mathbf{e}\mathbf{b}({\mathfrak{s}\mathfrak{p}}_6)$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\mathbf{F}\mathbf{u}\mathbf{n}\mathbf{d}\mathbf{R}\mathbf{e}\mathbf{p}(U_q({\mathfrak{s}\mathfrak{p}}_6))$ of quantum ${\mathfrak{s}\mathfrak{p}}_6$ representations generated by the fundamental representations, for generic values of the parameter $q$. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $$\mathbf{W}\mathbf{e}\mathbf{b}$(\mathfrak{sp}_6) \rightarrow $\mathbf{F}\mathbf{u}\mathbf{n}\mathbf{d}\mathbf{R}\mathbf{e}\mathbf{p}$(U_q(\mathfrak{sp}_6))$,thatall$$\mathrm{H}\mathrm{o}\mathrm{m}$$-spacesin$$\mathbf{W}\mathbf{e}\mathbf{b}$(\mathfrak{sp}_6)$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\mathbf{W}\mathbf{e}\mathbf{b}({\mathfrak{s}\mathfrak{p}}_6)$ is $1$-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum ${\mathfrak{s}\mathfrak{p}}_6$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.