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Smooth threefold flops with irreducible centres are classified by the lengthinvariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katzand Morrison identifies 6 possible partial resolutions of Kleiniansingularities that can occur as generic hyperplane sections, and thesimultaneous resolutions associated to such a partial resolution produce theuniversal flop of length $l$.  In this paper we translate these ideas into noncommutative algebra. Weintroduce the universal flopping algebra of length $l$ from which the universalflop of length $l$ can be recovered by a moduli construction, and we presenteach of these algebras as the path algebra of a quiver with relations. Thisexplicit realisation can then be used to construct examples of NCCRs associatedthreefold flops of any length as quiver with relations defined bysuperpotentials, to recover the matrix factorisation description of theuniversal flop conjectured by Curto and Morrison, and to realise examples ofcontraction algebras.

The length classification of threefold flops via noncommutative algebras