  This paper begins with a survey of some applications of Khovanov homology to
low-dimensional topology, with an eye toward extending these results to
$\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance
obstruction from Khovanov homology to $\mathfrak{sl}(n)$ homology for $n \geq
2$, including the universal $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$ homology
theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number
coming from Khovanov homology and relying on spectral sequence arguments, we
produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral
sequences also provide lower bounds on Turaev genus.
