  The Ryu-Takayanagi prescription can be cast in terms of a set of microscopic
threads that help visualize holographic entanglement in terms of distillation
of EPR pairs. While this framework has been exploited for regions with a high
degree of symmetry, we take the first steps towards understanding general
entangling regions, focusing on AdS$_4$. Inspired by simple constructions
achieved for the case of disks and the half-plane, we reformulate bit threads
in terms of a magnetic-like field generated by a current flowing through the
boundary of the entangling region. The construction is possible for these
highly symmetric settings, leading us to a modified Biot-Savart law in curved
space that fully characterizes the entanglement structure of the state. For
general entangling regions, the prescription breaks down as the corresponding
modular Hamiltonians become inherently nonlocal. We develop a formalism for
general shape deformations and derive a flow equation that accounts for these
effects as a systematic expansion. We solve this equation for a complete set of
small deformations and show that the structure of the expansion explicitly
codifies the expected nonlocalities. Our findings are consistent with numerical
results existing in the literature, and shed light on the fundamental nature of
quantum entanglement as a nonlocal phenomenon.
