  We give a combinatorial model for r-spin surfaces with parametrised boundary
based on Novak (2015). The r-spin structure is encoded in terms of
$\mathbb{Z}_r$-valued indices assigned to the edges of a polygonal
decomposition. This combinatorial model is designed for our state sum
construction of two-dimensional topological field theories on r-spin surfaces.
We show that an example of such a topological field theory computes the
Arf-invariant of an r-spin surface as introduced in Geiges, Gonzalo (2012) and
Randal-Williams (2014). This implies in particular that the r-spin
Arf-invariant is constant on orbits of the mapping class group, providing an
alternative proof of that fact.
