  Symmetry breaking can produce ``Alice'' strings, which alter scattered
charges and carry monopole number and charge when twisted into loops. Alice
behavior arises algebraically, when a string's untraced Wilson loop obstructs
unbroken symmetries -- a fragile criterion. We give a topological criterion,
compelling Alice behavior or deforming it away. Our criterion, that \pi_o(H)
acts nontrivially on \pi_1(H), links topologically Alice strings to topological
monopoles. We twist Alice loops to form monopoles, and find nematic and He-3A
Alice strings are topologically Alice, carrying fundamental monopole charge
when twisted into loops.
