  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.

Quantum Algebra