  We study the combinatorics of crystal graphs given by highest weight
representations of types $A_{n}, B_{n}, C_{n}$, and $D_{n}$, uncovering new
relations that exist among crystal operators. Much structure in these graphs
has been revealed by local relations given by Stembridge and Sternberg.
However, there exist relations among crystal operators that are not implied by
Stembridge or Sternberg relations. Viewing crystal graphs as edge colored
posets, we use poset topology to study them. Using the lexicographic discrete
Morse functions of Babson and Hersh, we relate the M\"obius function of a given
interval in a crystal poset of simply laced or doubly laced type to the types
of relations that can occur among crystal operators within this interval.
  For a crystal of a highest weight representation of finite classical Cartan
type, we show that whenever there exists an interval whose M\"obius function is
not equal to -1, 0, or 1, there must be a relation among crystal operators
within this interval not implied by Stembridge or Sternberg relations. As an
example of an application, this yields relations among crystal operators in
type $C_{n}$ that were not previously known. Additionally, by studying the
structure of Sternberg relations in the doubly laced case, we prove that
crystals of highest weight representations of types $B_{2}$ and $C_{2}$ are not
lattices.
