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The topological quantum number Q of a superconducting or chiral insulatingwire counts the number of stable bound states at the end points. We determine Qfrom the matrix r of reflection amplitudes from one of the ends, generalizingthe known result in the absence of time-reversal and chiral symmetry to allfive topologically nontrivial symmetry classes. The formula takes the form ofthe determinant, Pfaffian, or matrix signature of r, depending on whether r isa real matrix, a real antisymmetric matrix, or a Hermitian matrix. We applythis formula to calculate the topological quantum number of N coupled dimerizedpolymer chains, including the effects of disorder in the hopping constants. Thescattering theory relates a topological phase transition to a conductance peak,of quantized height and with a universal (symmetry class independent) lineshape. Two peaks which merge are annihilated in the superconducting symmetryclasses, while they reinforce each other in the chiral symmetry classes.

Scattering formula for the topological quantum number of a disordered multi-mode wire