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Onsite gain-loss induced topological braiding principles of non-Hermitian
energy bands is theoretically formulated in multiband lattice models with
Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss
parameter is tuned across exceptional point degeneracies. Laboratory realizable
effective-Hamiltonians are proposed to realize braid groups $\mathbb{B}_2$ and
$\mathbb{B}_3$ of two and three bands respectively. While $\mathbb{B}_2$ is
trivially Abelian, the group $\mathbb{B}_3$ features non-Abelian braiding and
energy permutation. Phase diagrams with respect to lattice parameters to
realize braid group generators and their non-commutativity are shown. The
proposed theory is conducive to synthesize exceptional materials for
applications in topological quantum photonic computation and information
processing.