  Smooth threefold flops with irreducible centres are classified by the length
invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz
and Morrison identifies 6 possible partial resolutions of Kleinian
singularities that can occur as generic hyperplane sections, and the
simultaneous resolutions associated to such a partial resolution produce the
universal flop of length $l$.
  In this paper we translate these ideas into noncommutative algebra. We
introduce the universal flopping algebra of length $l$ from which the universal
flop of length $l$ can be recovered by a moduli construction, and we present
each of these algebras as the path algebra of a quiver with relations. This
explicit realisation can then be used to construct examples of NCCRs associated
threefold flops of any length as quiver with relations defined by
superpotentials, to recover the matrix factorisation description of the
universal flop conjectured by Curto and Morrison, and to realise examples of
contraction algebras.
