  Turaev Viro invariants are amongst the most powerful tools to distinguish
3-manifolds: They are implemented in mathematical software, and allow practical
computations. The invariants can be computed purely combinatorially by
enumerating colourings on the edges of a triangulation T.
  These edge colourings can be interpreted as embeddings of surfaces in T. We
give a characterisation of how these embedded surfaces intersect with the
tetrahedra of T. This is done by characterising isotopy classes of simple
closed loops in the 3-punctured disk. As a direct result we obtain a new system
of coordinates for edge colourings which allows for simpler definitions of the
tetrahedron weights incorporated in the Turaev-Viro invariants.
  Moreover, building on a detailed analysis of the colourings, as well as
classical work due to Kirby and Melvin, Matveev, and others, we show that
considering a much smaller set of colourings suffices to compute Turaev-Viro
invariants in certain significant cases. This results in a substantial
improvement of running times to compute the invariants, reducing the number of
colourings to consider by a factor of $2^n$. In addition, we present an
algorithm to compute Turaev-Viro invariants of degree four -- a problem known
to be #P-hard -- which capitalises on the combinatorial structure of the input.
  The improved algorithms are shown to be optimal in the following sense: There
exist triangulations admitting all colourings the algorithms consider.
Furthermore, we demonstrate that our new algorithms to compute Turaev-Viro
invariants are able to distinguish the majority of $\mathbb{Z}$-homology
spheres with complexity up to $11$ in $O(2^n)$ operations in $\mathbb{Q}$.
