We study webs in quantum type C, focusing on the rank three case. We define a linear pivotal category $\textbf {Web}(\mathfrak {sp}_{6})$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ of quantum $\mathfrak {sp}_{6}$ 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 $\textbf {Web}(\mathfrak {sp}_{6}) \rightarrow \textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ , that all $\textrm {Hom}$ -spaces in $\textbf {Web}(\mathfrak {sp}_{6})$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\textbf {Web}(\mathfrak {sp}_{6})$ is one-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 $\mathfrak {sp}_{6}$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.

我们研究量子类型C的网络，重点关注等级 3 的情况。我们通过生成器和关系图形化地定义了一个线性关键类别 $\textbf {Web}(\mathfrak {sp}_{6})$ ，并推测它等价于类别 $\textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ 的量子 $\mathfrak {sp}_{6}$ 表示由基本表示生成，用于参数q的通用值。我们证明了许多支持这个猜想的结果，最值得注意的是存在一个完整的，本质上是满射的函子 $\textbf {Web}(\mathfrak {sp}_{6}) \rightarrow \textbf {FundRep}(U_q( \mathfrak {sp}_{6}))$ ，所有 $\textrm {Hom}$ - $\textbf {Web}(\mathfrak {sp}_{6})$ 中的空间是有限维的，并且 $\textbf {Web}(\mathfrak {sp}_{ 6})$ 是一维的。后者对应于可以使用局部关系将所有闭合网络评估为标量的陈述；因此，我们获得了量子 $\mathfrak {sp}_{6}$ 链接不变量的新方法，类似于琼斯多项式的考夫曼括号描述。