  We give a scheme for interpreting shaded tangles as quantum circuits, with
the property that if two shaded tangles are ambient isotopic, their
corresponding computational effects are identical. We analyze 11 known quantum
procedures in this way -- including entanglement manipulation, error correction
and teleportation -- and in each case present a fully-topological formal
verification, yielding generalized procedures in some cases. We also use our
methods to identify 2 new procedures, for topological state transfer and
quantum error correction. Our formalism yields in some cases significant new
insight into how the procedures work, including a description of quantum
entanglement arising from topological entanglement of strands, and a
description of quantum error correction where errors are `trapped by bubbles'
and removed from the shaded tangle.
