  We consider the existence and spectral stability of static multi-kink
structures in the discrete sine-Gordon equation, as a representative example of
the family of discrete Klein-Gordon models. The multi-kinks are constructed
using Lin's method from an alternating sequence of well-separated kink and
antikink solutions. We then locate the point spectrum associated with these
multi-kink solutions by reducing the spectral problem to a matrix equation. For
an $m$-structure multi-kink, there will be $m$ eigenvalues in the point
spectrum near each eigenvalue of the primary kink, and, as long as the spectrum
of the primary kink is imaginary, the spectrum of the multi-kink will be as
well. We obtain analytic expressions for the eigenvalues of a multi-kink in
terms of the eigenvalues and corresponding eigenfunctions of the primary kink,
and these are in very good agreement with numerical results. We also perform
numerical time-stepping experiments on perturbations of multi-kinks, and the
outcomes of these simulations are interpreted using the spectral results.
