  The cutoff phenomenon describes the case when an abrupt transition occurs in
the convergence of a Markov chain to its equilibrium measure. There are various
metrics which can be used to measure the distance to equilibrium, each of which
corresponding to a different notion of cutoff. The most commonly used are the
total-variation and the separation distances. In this note we prove that the
cutoff for these two distances are not equivalent by constructing several
counterexamples which display cutoff in total-variation but not in separation
and with the opposite behavior, including lazy simple random walk on a sequence
of uniformly bounded degree expander graphs. These examples give a negative
answer to a question of Ding, Lubetzky and Peres.
