  In previous work with Scullard, we defined a graph polynomial P_B(q,T) that
gives access to the critical temperature T_c of the q-state Potts model on a
general two-dimensional lattice L. It depends on a basis B, containing n x m
unit cells of L, and the relevant root of P_B(q,T) was observed to converge
quickly to T_c in the limit n,m to infinity. Moreover, in exactly solvable
cases there is no finite-size dependence at all.
  We reformulate this method as an eigenvalue problem within the periodic
Temperley-Lieb algebra. This corresponds to taking m to infinity first, so the
bases B are semi-infinite cylinders of circumference n. The limit implies
faster convergence in n, while maintaining the n-independence in exactly
solvable cases. In this setup, T_c(n) is determined by equating the largest
eigenvalues of two topologically distinct sectors of the transfer matrix.
Crucially, these two sectors determine the same critical exponent in the
continuum limit, and the observed fast convergence is thus corroborated by
results of conformal field theory. We obtain similar results for the dense and
dilute phases of the O(N) loop model, using now a transfer matrix within the
dilute periodic Temperley-Lieb algebra.
  The eigenvalue formulation allows us to double the size n for which T_c(n)
can be obtained, using the same computational effort. We study in details three
significant cases: (i) bond percolation on the kagome lattice, up to n = 14;
(ii) site percolation on the square lattice, to n = 21; and (iii) self-avoiding
polygons on the square lattice, to n = 19. Convergence properties of T_c(n) and
extrapolation schemes are studied in details for the first two cases. This
leads to rather accurate values for the percolation thresholds: p_c =
0.524404999167439(4) for bond percolation on the kagome lattice, and p_c =
0.59274605079210(2) for site percolation on the square lattice.
