  This paper provides both a detailed study of color-dependence of link
homologies, as realized in physics as certain spaces of BPS states, and a broad
study of the behavior of BPS states in general. We consider how the spectrum of
BPS states varies as continuous parameters of a theory are perturbed. This
question can be posed in a wide variety of physical contexts, and we answer it
by proposing that the relationship between unperturbed and perturbed BPS
spectra is described by a spectral sequence. These general considerations unify
previous applications of spectral sequence techniques to physics, and explain
from a physical standpoint the appearance of many spectral sequences relating
various link homology theories to one another. We also study structural
properties of colored HOMFLY homology for links and evaluate Poincare
polynomials in numerous examples. Among these structural properties is a novel
"sliding" property, which can be explained by using (refined) modular S-matrix.
This leads to the identification of modular transformations in Chern-Simons
theory and 3d N=2 theory via the 3d/3d correspondence. Lastly, we introduce the
notion of associated varieties as classical limits of recursion relations of
colored superpolynomials of links, and study their properties.
