  We derive parametric integral representations for the general $n$-point
function of scalar operators in momentum-space conformal field theory.
Recently, this was shown to be expressible as a generalised Feynman integral
with the topology of an $(n-1)$-simplex, featuring an arbitrary function of
momentum-space cross ratios. Here, we show all graph polynomials for this
integral can be expressed in terms of the first and second minors of the
Laplacian matrix for the simplex. Computing the effective resistance between
nodes of the corresponding electrical network, an inverse parametrisation is
found in terms of the determinant and first minors of the Cayley-Menger matrix.
These parametrisations reveal new families of weight-shifting operators,
expressible as determinants, that connect $n$-point functions in spacetime
dimensions differing by two. Moreover, the action of all previously known
weight-shifting operators preserving the spacetime dimension is manifest.
Finally, the new parametric representations enable the validity of the
conformal Ward identities to be established directly, without recourse to
recursion in the number of points.
