  We establish new results on the spectra and pseudo-spectra of tridiagonal
$k$-Toeplitz operators and matrices. In particular, we prove the connection
between the winding number of the eigenvalues of the symbol function and the
exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our
results elucidate the topological origin of the non-Hermitian skin effect in
general one-dimensional polymer systems of subwavelength resonators with
imaginary gauge potentials, proving the observation and conjecture in
arXiv:2307.13551. We also numerically verify our theory for these systems.
