  The preceding paper constructed tangle machines as diagrammatic models, and
illustrated their utility with a number of examples. The information content of
a tangle machine is contained in characteristic quantities associated to
equivalence classes of tangle machines, which are called invariants. This paper
constructs invariants of tangle machines. Chief among these are the prime
factorizations of a machine, which are essentially unique. This is proven using
low dimensional topology, through representing a colour-suppressed machine as a
diagram for a network of jointly embedded spheres and intervals in 4-space. The
complexity of a tangle machine is defined as its number of prime factors.
