  We show that the critical Kac-Ward operator on isoradial graphs acts in a
certain sense as the operator of s-holomorphicity, and we identify the
fermionic observable for the spin Ising model as the inverse of this operator.
This result is partially a consequence of a more general observation that the
inverse Kac-Ward operator for any planar graph is given by what we call a
fermionic generating function. Furthermore, using bounds for the spectral
radius and operator norm of the Kac-Ward transition matrix, we provide a
general picture of the non-backtracking walk representation of the critical and
supercritical inverse Kac-Ward operators on isoradial graphs.
