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In this paper, I compute the inhomogeneous (multi-probability) bond criticalsurfaces for the (4,6,12) and (3^4,6) lattices using the linearityapproximation described in (Scullard and Ziff, J. Stat. Mech. P03021),implemented as a branching process of lattices. I find the estimates for thebond percolation thresholds, p_c(4,6,12)=0.69377849... andp_c(3^4,6)=0.43437077..., compared with Parviainen's numerical results of p_c\approx 0.69373383 and p_c \approx 0.43430621 . These deviations are of theorder 10^{-5}, as is standard for this method, although they are outsideParviainen's typical standard error of 10^{-7}. Deriving thresholds in this wayfor a given lattice leads to a polynomial with integer coefficients, the rootin [0,1] of which gives the estimate for the bond threshold. I show how themethod can be refined, leading to a sequence of higher order polynomials makingpredictions that likely converge to the exact answer. Finally, I discuss howthis fact hints that for certain graphs, such as the kagome lattice, the exactbond threshold may not be the root of any polynomial with integer coefficients.

Polynomial sequences for bond percolation critical thresholds