| A threefold flopping contraction is a birational morphism X to Y of complex threefolds that contracts finitely-many single smooth rational curves C, with the canonical class K_X trivial on each one. Threefold flops are not classified, even in the case X smooth, but many invariants are known. In particular, the general section of the image of a flop in Y is an A1, D4, E6, E7 or E8 singularity (and the exceptional curve extracted on X is known in each case). The A1 case was classified in the 1980s (by Reid) and examples in the case D4 known (by Laufer). Recently Joe Karmazyn showed that flops of all types really do exist. With Michael Wemyss, I construct explicit cases of flops of all types. A first corollary of this is that the Gopukumar-Vafa invariants (which count curves in small deformations) are not sufficient to disinguish flops. Instead, the Contraction algebra (of Donovan-Wemyss) distinguishes our examples. |