We characterize all equivariant odd  spectral triples on the quantum $SU(2)$ 
group having a nontrivial Chern character.  It is shown that the dimension of 
an equivariant spectral triple is at least three, and there does exist
a 3-summable equivariant spectral triple. We also show that given any 
odd spectral triple, there is an odd equivariant spectral triple that induces 
the same element in $K^1$.