We introduce a topological flux function to quantify the topology of magnetic braids: non-zero, line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, and measures the average poloidal magnetic flux around any given field line, or the average pairwise crossing number between a given field line and all others. Moreover, its integral over the cross-section yields the relative

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... tic helicity. Using the fact that the flux function is also an action in the Hamiltonian formulation of the field line equations, we prove that it uniquely characterizes the field line mapping and hence the magnetic topology.