A blade with the handle curving away from the cutting edge, drawn in two strokes. Standing alone it stays 刀; squeezed onto the right of a character it flattens into 刂, which is why 别, 到 and 前 all end in a pair of vertical strokes that used to be a knife.
刀 was also money: the Warring States states of Qi and Yan cast bronze coins shaped like knives, 刀币. Today 一刀切 ('cut it all with one knife') is the standard criticism of a blunt one-size-fits-all policy, and 刀子嘴豆腐心 describes someone with a knife for a mouth and tofu for a heart.
A curved KNIFE — the short stroke is the handle, the long one the blade.
快 means both 'fast' and 'sharp'. A sharp knife is a 快刀, because it goes through things quickly.