  Braided vector fields on spatial subdomains homeomorphic to the cylinder play
a crucial role in applications such as solar and plasma physics, relativistic
astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often
the vector field's topology -- the entanglement of its field lines -- is
non-trivial, and can play a significant role in the vector field's evolution.
We present a complete topological characterisation of such vector fields (up to
isotopy) using a quantity called field line winding. This measures the
entanglement of each field line with all other field lines of the vector field,
and may be defined for an arbitrary tubular subdomain by prescribing a
minimally distorted coordinate system. We propose how to define such
coordinates, and prove that the resulting field line winding distribution
uniquely classifies the topology of a braided vector field. The field line
winding is similar to the field line helicity considered previously for
magnetic (solenoidal) fields, but is a more fundamental measure of the field
line topology because it does not conflate linking information with field
strength.
