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This work completes the classification of the cubic vertices for arbitraryspin massless bosons in three dimensions started in a previous companion paperby constructing parity-odd vertices. Similarly to the parity-even case, thereis a unique parity-odd vertex for any given triple $s_1\geq s_2\geq s_3\geq 2$of massless bosons if the triangle inequalities are satisfied ($s_1<s_2+s_3$)and none otherwise. These vertices involve two (three) derivatives for odd(even) values of the sum $s_1+s_2+s_3$. A non-trivial relation betweenparity-even and parity-odd vertices is found. Similarly to the parity-evencase, the scalar and Maxwell matter can couple to higher spins through currentcouplings with higher derivatives. We comment on possible lessons for 2d CFT.We also derive both parity-even and parity-odd vertices with Chern-Simonsfields and comment on the analogous classification in two dimensions.

Cubic interactions of massless bosonic fields in three dimensions II: Parity-odd and Chern-Simons vertices