  Electrical and thermal transport on a fractional quantum Hall edge are
determined by topological quantities inherited from the corresponding bulk
state. While electrical transport is the standard method for studying edges,
thermal transport appears more challenging. Here, we show that the shot noise
generated on the edge provides a fully electrical method to probe the edge
structure. In the incoherent regime, the noise falls into three topologically
distinct universality classes: charge transport is always ballistic while
thermal transport is either ballistic, diffusive, or "antiballistic".
Correspondingly, the noise either vanishes, decays algebraically or is constant
up to exponentially small corrections in the edge length.
