  By generalising the notion of a unimodular sequence, we create an expression
for the winding number of certain ordered sets of lattice points. Since the
winding number of the vertices of a Fano polygon is necessarily one, we use
this expression as a restriction to classify all Fano polygons without
T-singularities and whose basket of residual singularities is of the form
$\left\{ \frac{1}{r}(1,s_{1}), \frac{1}{r}(1,s_{2}), \ldots,
\frac{1}{r}(1,s_{k}) \right\}$ for $k,r \in \mathbb{Z}_{>0}$, and $1 \leq s_{i}
< r$ is coprime to $r$.
