Predictions of bond percolation thresholds for the kagom\'e and Archimedean (3,12²) lattices

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.