  The smallest known example of a family of modular categories that is not
determined by its modular data are the rank 49 categories
$\mathcal{Z}(\text{Vec}_G^{\omega})$ for $G=\mathbb{Z}_{11} \rtimes
\mathbb{Z}_{5}$. However, these categories can be distinguished with the
addition of a matrix of invariants called the $W$-matrix that contains
intrinsic information about punctured $S$-matrices. Here we show that it is a
common occurrence for knot and link invariants to carry more information than
the modular data. We present the results of a systematic investigation of the
invariants for small knots and links. We find many small knots and links that
are complete invariants of the $\mathcal{Z}(\text{Vec}_G^{\omega})$ when
$G=\mathbb{Z}_{11} \rtimes \mathbb{Z}_{5}$, including the $5_2$ knot.
