Jones polynomial of a split union ($V_{L\bigsqcup J}(t)=-(t^{1/2}+t^{-1/2})V_L(t)V_J(t)$)

Placing the two link diagrams in disjoint disks keeps all state circles separate. The reduced Kauffman bracket therefore acquires one extra loop factor $-A^2-A^{-2}$. Correcting writhe of a link diagram and setting $t=A^{-4}$ gives the displayed formula. It concerns a split link, not an arbitrary union of linked components.