Admissibility and the $C_2$ Spider

A tensor category is multiplicity-free if for any objects $A,B,C$ we have that $\mathrm{Hom}(A\otimes B\otimes C,\mathbb{C})$ is either $0$ or $1$ dimensional. It is known that $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ is not multiplicty-free. We find a full subcategory of $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ which is multiplicty-free. A description of the dimension of these $\mathrm{Hom}$ spaces is given for this subcategory, including when $q$ is a root of unity. The methods used arise from the description, given by Kuperberg, of $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ as a spider. The main tool is the recursive definition of clasps given by Kim. In particular, we provide an appropriate notion of admissibility when looking at the $\mathrm{Sp}(4)_k$ ribbon graph invariants with restricted edge labels.