  We study Markov chains which model genome rearrangements. These models are
useful for studying the equilibrium distribution of chromosomal lengths, and
are used in methods for estimating genomic distances. The primary Markov chain
studied in this paper is the top-swap Markov chain. The top-swap chain is a
card-shuffling process with $n$ cards divided over $k$ decks, where the cards
are ordered within each deck. A transition consists of choosing a random pair
of cards, and if the cards lie in different decks, we cut each deck at the
chosen card and exchange the tops of the two decks. We prove precise bounds on
the relaxation time (inverse spectral gap) of the top-swap chain. In
particular, we prove the relaxation time is $\Theta(n+k)$. This resolves an
open question of Durrett.
