  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 strings obstruct 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
topologically Alice loops to form monopoles. We show that Alice strings of
condensed matter systems (nematic liquid crystals, helium 3A, and related
non-chiral Bose condensates and amorphous chiral superconductors) are
topologically Alice, and support fundamental monopole charge when twisted into
loops. Thus they might be observed indirectly, not as strings, but as loop-like
point defects. We describe other models, showing Alice strings failing our
topological criterion; and twisted Alice loops supporting deposited, but not
fundamental, monopole number.
