  We give a conditional derivation of the inhomogeneous critical percolation
manifold of the bow-tie lattice with five different probabilities, a problem
that does not appear at first to fall into any known solvable class. Although
our argument is mathematically rigorous only on a region of the manifold, we
conjecture that the formula is correct over its entire domain, and we provide a
non-rigorous argument for this that employs the negative probability regime of
the triangular lattice critical surface. We discuss how the rigorous portion of
our result substantially broadens the range of lattices in the solvable class
to include certain inhomogeneous and asymmetric bow-tie lattices, and that, if
it could be put on a firm foundation, the negative probability portion of our
method would extend this class to many further systems, including F.Y. Wu's
checkerboard formula for the square lattice. We conclude by showing that this
latter problem can in fact be proved using a recent result of Grimmett and
Manolescu for isoradial graphs, lending strong evidence in favour of our other
conjectured results.
