  We study the percolation critical surface of the kagome lattice in which each
triangle is allowed an arbitrary connectivity. Using the method of critical
polynomials, we find points along this critical surface to high precision. This
kagome hypergraph contains many unsolved problems as special cases, including
bond percolation on the kagome and $(3,12^2)$ lattices, and site percolation on
the hexagonal, or honeycomb, lattice, as well as a single point for which there
is an exact solution. We are able to compute enough points along the critical
surface to find a very accurate fit, essentially a Taylor series about the
exact point, that allows estimations of the critical point of any system that
lies on the surface to precision rivaling Monte Carlo and traditional
techniques of similar accuracy. We find also that this system sheds light on
some of the surprising aspects of the method of critical polynomials, such as
why it is so accurate for certain problems, like the kagome and $(3,12^2)$
lattices. The bond percolation critical points of these lattices can be found
to 17 and 18 digits, respectively, because they are in close proximity, in a
sense that can be made quantitative, to the exact point on the critical
surface. We also discuss in detail a parallel implementation of the method
which we use here for a few calculations.
